
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00005103/00001
Material Information
 Title:
 Computational approaches in graph theory
 Creator:
 Brandt, Axel Thomas ( author )
 Language:
 English
 Physical Description:
 1 electronic file (101 pages). : ;
Thesis/Dissertation Information
 Degree:
 Doctor of philosophy
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Mathematical and Statistical Sciences, CU Denver
 Degree Disciplines:
 Applied mathematics
Subjects
 Subjects / Keywords:
 Graph theory  Data processing ( lcsh )
Bipartite graphs ( lcsh ) Bipartite graphs ( fast ) Graph theory  Data processing ( fast )
 Genre:
 nonfiction ( marcgt )
Notes
 Review:
 We consider three problems in graph theory and intertwine advanced mathematical theory with sophisticated computational methods to obtain solutions. The first two problems utilize the theory of flag algebras in combination with semidefinite programming to obtain results in extremal graph theory. The third problem uses computational tools to facilitate a novel application of the Combinatorial Nullstellensatz within proofs using the discharging method.
 Review:
 First, we investigate the minimum edge density between parts of a fourpartite graph that guarantees the existence of a triangle. Our approach relies heavily upon developing a flag algebra model that views the problem as coloring the edges of a complete graph.
 Review:
 Next, we pursue a construction that obtains the maximum number of induced kcycles in a graph of fixed order. In 1975, Pippenger and Golumbic conjectured that the balanced iterated blowup of a cycle is the extremal construction for all induced cycles on at least 5 vertices. This conjecture was affirmed earlier this year for induced 5cycles on graphs of specific sizes. Using a similar approach, our results affirm this conjecture for induced 6cycles on graphs of order 6m for mâˆˆℕ and provide a second piece of evidence that the conjecture may be true. Initial computational experiments indicate that we may be able to extend our methods to affirm this conjecture for k=7,8 on graphs of certain order.
 Review:
 Finally, we seek to determine the "lucky number" of graphs. For any vertex labeling ℓ:V(G)â†’ℝ we say that ℓ is lucky if the neighborhoods for every pair of adjacent vertices have distinct sums of vertex labels. The least integer k for which a graph has a lucky labeling using labels from {1,â€¦,k} is its lucky number. In 2009, CzerwinÌÂski, Grytczuk, and ZÌÂ‡elazny conjectured that the lucky number of a graph is at most its chromatic number. This conjecture has only been confirmed for a few graph classes. Our results provide new and improved bounds for multiple families of planar graphs. In doing so, we confirm this conjecture for nonbipartite planar graphs with girth at least 26.
 Thesis:
 Thesis (M.S.)University of Colorado Denver.
 Bibliography:
 Includes bibliographic references.
 System Details:
 System requirements: Internet connectivity.
 Statement of Responsibility:
 by Axel Thomas Brandt.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 957593542 ( OCLC )
ocn957593542

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COMPUTATIONAL APPROACHES IN GRAPH THEORY
by
AXEL THOMAS BRANDT
B.S., Mathematics, Ohio Northern University, 2010
M.S., Mathematics, Miami University, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2016
This thesis for the Doctor of Philosophy degree by
Axel Thomas Brandt
has been approved for the
Department of Mathematical and Statistical Sciences
by
Florian Pfender, Advisor
Michael Ferrara, Chair
Paul Florn
Michael Jacobson
Sogol Jahanbekam
April 28, 2016
11
Brandt, Axel Thomas (Ph.D., Applied Mathematics)
Computational Approaches in Graph Theory
Thesis directed by Associate Professor Florian Pfender
ABSTRACT
We consider three problems in graph theory and intertwine advanced mathe
matical theory with sophisticated computational methods to obtain solutions. The
first two problems utilize the theory of flag algebras in combination with semidehnite
programming to obtain results in extremal graph theory. The third problem uses com
putational tools to facilitate a novel application of the Combinatorial Nullstellensatz
within proofs using the discharging method.
First, we investigate the minimum edge density between parts of a fourpartite
graph that guarantees the existence of a triangle. Our approach relies heavily upon
developing a flag algebra model that views the problem as coloring the edges of a
complete graph.
Next, we pursue a construction that obtains the maximum number of induced
fccycles in a graph of fixed order. In 1975, Pippenger and Golumbic conjectured that
the balanced iterated blowup of a cycle is the extremal construction for all induced
cycles on at least 5 vertices. This conjecture was affirmed earlier this year for induced
5cycles on graphs of specific sizes. Using a similar approach, our results affirm this
conjecture for induced 6cycles on graphs of order 6m for m E N and provide a second
piece of evidence that the conjecture may be true. Initial computational experiments
indicate that we may be able to extend our methods to affirm this conjecture for
k = 7, 8 on graphs of certain order.
Finally, we seek to determine the "lucky number" of graphs. For any vertex
labeling t : V(G) 4lwe say that t is lucky if the neighborhoods for every pair of
iii
adjacent vertices have distinct sums of vertex labels. The least integer k for which a
graph has a lucky labeling using labels from {1,..., k} is its lucky number. In 2009,
Czerwinski, Grytczuk, and Zelazny conjectured that the lucky number of a graph is
at most its chromatic number. This conjecture has only been confirmed for a few
graph classes. Our results provide new and improved bounds for multiple families of
planar graphs. In doing so, we confirm this conjecture for nonbipartite planar graphs
with girth at least 26.
The form and content of this abstract are approved. I recommend its publication.
Approved: Florian Pfender
IV
For my future students
v
ACKNOWLEDGMENT
First and foremost, thank you to my wife Kimberly for your love, understanding,
and support over the years and particularly throughout my graduate career; without
it I would not be who I am today.
I would like to express an immense amount of gratitude to Flo, my advisor, for
his guidance through my doctoral studies, and a massive amount of appreciation to
Mike, my committee chair and seemingly unofficial advisor, for his active mentorship
in my growth as both a mathematician and an educator. For their extensive efforts
in my development I am deeply indebted and inexplicably thankful.
To Brent, my office mate, thank you for putting up with me on a daily basis for
the past four years. To Jenny, Cathy, Tim, and Phil, former graduate students, thank
you for your encouragement and example. To Flenc and Eric, also former graduate
students, thank you for pushing me to learn BTgX and Tikz early on in my graduate
studies. To Devon, a fellow graduate student, thank you for your patience with my
many naive optimization questions.
Thank you to my coauthors for collaborating with me, and particularly to Flo,
Bernard, Sogol, and Jenny for collaborations that led to results contained in this
manuscript. Additionally, thank you to the many American taxpayers without whom
the following NSF grants that partially funded my research would not have existed:
DGE0742434, DMS1427526, DMS1500662, and IUSE1539692.
Furthermore, thank you to the many teachers who have fostered both my love
of mathematics and passion for education over the years: Dan, Mihai, Sandy, Don,
Kristen, Laura, Dave, and John. In particular, a huge thank you to Tao, my masters
advisor, for encouraging me to pursue doctoral studies.
Finally, thank you to my family and friends for their support and encouragement
over the years. In particular, thank you to Bill and Jan for convincing me to pursue
doctoral studies and thank you so much to mom, Bruce, and Elbe for proofreading
this document.
vi
TABLE OF CONTENTS
LIST OF FIGURES........................................................ viii
COMPUTATIONS IN EXTREMAL GRAPH THEORY..................................... 1
1. An Introduction to Extremal Graph Theory...................... 2
1.1 Stability Method........................................... 4
1.2 Flag Algebras.............................................. 5
1.3 Convex Optimization........................................ 9
1.4 Obtaining Extremal Examples .............................. 11
2. Density Bounds in Multipartite Graphs......................... 16
2.1 Triangles in Multipartite Graphs.......................... 16
2.2 A Proposed Extremal Construction.......................... 17
2.3 A Flag Algebra Model...................................... 19
2.4 Multipartite Result....................................... 20
3. Induced Cycles.................................................. 28
3.1 Induced 5Cycles.......................................... 30
3.2 Induced 6Cycles.......................................... 32
3.2.1 Top Level Structure of the Extremal Graph ........ 33
3.2.2 Iterative Structure of the Extremal Graph......... 53
COMPUTATIONS IN GRAPH COLORING........................................... 56
4. An Introduction to Graph Coloring .............................. 57
4.1 The Four Color Theorem and the Discharging Method .... 58
4.2 List Coloring and the Combinatorial Nullstellensatz....... 62
5. Lucky Labeling ................................................. 65
5.1 Notation and Tools........................................ 68
5.2 Reducible Configurations ................................. 70
5.3 Lucky Labeling Results ................................... 77
REFERENCES............................................................... 87
vii
LIST OF FIGURES
Figure
2.1 The general structure of a conjectured extremal graph for d\.............. 17
2.2 Flag algebra constraints for vertexlabeled graph modeling a 4partite,
TGfree graph............................................................ 19
2.3 Comparing the number of flags on small vertex sets........................ 20
2.4 Flag algebra constraints for 3edgecolored complete graph modeling a
balanced, 4partite, K3free graph....................................... 21
2.5 Flag algebra constraints for 4edgecolored complete graph modeling a
12partite, K3free graph................................................ 27
3.1 Progression of obtaining a balanced, iterated blowup of C4............... 29
3.2 The extremal graph for i(Ce), CqX......................................... 32
3.3 Graphs in C22 and C3...................................................... 33
3.4 The embeddings of an induced P5 with one funky edge in G[X0, X\]. . . 40
3.5 The embeddings of an induced P5 with funky edge in G[X0,X 1]........... 41
3.6 The embeddings of an induced P5 with funky edge in G[X0,Xs]............ 43
3.7 Induced C6s with all funky edges incident to a single vertex.............. 46
5.1 An 8reducible configuration.............................................. 72
5.2 A reducible configuration................................................. 73
5.3 Some 3reducible configurations........................................... 74
viii
COMPUTATIONS IN EXTREMAL GRAPH THEORY
Extremal graph theory is a broad area of research in graph theory. At its core, this
area of research seeks to identify and learn about threshold phenomena. The origi
nating problem in this area concerns maximizing the number of edges in a graph that
avoids a specified forbidden subgraph and identifying the graph or graphs that obtain
this maximum. In joint work with Florian Pfender and Bernard Lidicky, the theory
of flag algebras is used toward identifying thresholds and the associated graphs for
two problems. The author contributed to the first problem by identifying flag algebra
constraints that model the first problem and writing code for their implementation.
The author contributed to the second problem by identifying flags that were crucial
in obtaining useful computational bounds, and verifying many of the details within
stability arguments.
1
CHAPTER 1
AN INTRODUCTION TO EXTREMAL GRAPH THEORY
Extremal graph theory focuses on both the relations between various graph pa
rameters, such as order, size, minimum and maximum degree, and chromatic number,
and also the values of these parameters that guarantee certain graph properties. More
formally, given a property V and a parameter p for a family T of graphs, we seek
to determine the minimum or maximum value m for which every graph G G T with
p(G) > m or p(G) < m, respectively, has property V. Graphs G E T with p(G) = m
are called the extremal graphs. As a toy example, suppose we are searching for the
minimum m such that every graph G of order n with \E(G)\ > m implies that G
contains a cycle. The minimum such m is n, and the family of trees on n vertices
constitutes the extremal graphs for this problem. In this case, T is the family of all
graphs on n vertices, V is the property that a graph contains a cycle, p(G) represents
\E(G)\, and m = n.
This toy example can also be viewed as searching for the maximum m such that
every graph G of order n with if(G) < m implies that G is acyclic. This phrasing
casts our example as a forbidden subgraph problem. An Tfree graph is a graph that
does not contain any graph in a forbidden family of graphs T. For the forbidden
subgraph problem, the notation ex(n,tF) denotes the maximum number of edges in
an JMree graph of order n. Although Mantel [69] determined ex(n, A3) in 1907,
Turans [90] determination of ex(n, Kk) for all k > 3 is widely viewed as the birth
of modern extremal graph theory, which subsequently developed in large part due to
Paul Erdos extensive investigation of numerous extremal problems. Turans result
states that among all Ar+1free graphs on n vertices, the graph with the most edges is
obtained by the complete multipartite graph with r parts, each having either _yj or
~^~ vertices; this graph is denoted T(n,r) and is called the rpartite rpartite Turdn
graph on n vertices.
2
Counting the number of copies of a fixed graph H in a host graph G is a natural
extension of the problem of determining if H is a subgraph of G. For instance, Erdos
and Stone [38] guarantee that a large number of cliques appear in sufficiently large
graphs with a relatively small number of edges beyond what Turans theorem requires
to guarantee a single clique. Erdos and Simonovits [37] use the aforementioned result
to provide the proportion of edges in extremal graphs of order n as n > oo dependent
solely upon the maximum chromatic number of a graph in the forbidden family.
For much of this chapter we direct our attention to large graphs. In this setting,
rather than examining ex(n,T) for large n it is perhaps natural to examine the
proportion of the total number of possible edges in an extremal graph. The following is
a straightforward example of how this perspective can be more useful than comparing
the exact edge counts of various large graphs:
Proposition 1.1 For every family T,
ex{n,T)
is decreasing as n > oo.
Proof: Let H be an extremal Jfree graph on m vertices. Let n < m and G\,... ,Gt
be the () induced subgraphs of H on n vertices. Notice that each edge of H is in
C^l!) f the GiS. Thus,
ex(m, T)
m 2
n 2
f) <Â£Â£(&)
i=l
n 2
Since /(
m2
n2
m(m 1)
n(n 1)
(/ (2) > we ^ave
ex(m,iF) ex(n,T)
Going forward in this chapter, we first discuss the structural properties of graphs
having a parameter close to that of an extremal graph. Then, we discuss a very
recently introduced tool in extremal combinatorics, the theory of flag algebras, that we
will use to obtain extremal results in subsequent chapters. As implementation of the
3
flag algebra method relies on computational tools, the conclusion of this chapter will
provide an overview of the optimization theory underlying flag algebra computations
and an example thereof.
Before proceeding, it will be useful to introduce notation that describes the
asymptotic behavior of functions. Let n be an integer variable that tends to in
finity and let (f>(n) be a positive function and f(n) be an arbitrary function. It is
said that / G 0(f) provided that / < cf for some constant c and all values of
n. Similarly, / G o(f) provided that // f > 0, that is grows much faster than /.
Analogous statements can be made for functions of a continuous variable tending to
some limit.
1.1 Stability Method
When the structure of extremal graphs is described as stable, it means that graphs
that are nearly extremal (regarding the parameter in question) have a structure that
is close to that of the extremal graphs. The following example of stability is a result
of combined efforts of Erdos and Simonovits:
Theorem 1.2 (The First Stability Theorem, [33, 34, 82]) Let G be a graph on
n vertices and T be a family of forbidden graphs with min{x(G): G G J} =p + l.
For every e > 0, there exists 5 > 0 and ne such that if G is Tfree and if, for n > ne,
\E(G)\ > ex(n,T) 6n2,
then G can be obtained from, T(n,p) by changing at most tv2 edges.
In the context of maximizing edges in the forbidden subgraph problem, this theo
rem states that an Ffree graph with nearly ex(n, F) edges is structurally similar to
T(n, x(F) 1). By structurally similar, we mean that the nearly extremal graph can
be altered to obtain T(n,x(F) 1) by changing relatively few edges.
As the name suggests, there are more stability theorems; see the book by Bollobas
[14] or the survey by Simonovits [83]. Many stability results are utilized as a precursor
4
to proving graphs to be extremal. Our results in the upcoming chapters will use this
approach, which begins with a graph H whose parameter p for a property V exceeds
that of the proposed extremal graph G. We upper bound p(H) by p(G) +h for some
small 8 > 0 and then use the closeness of p(H) and p(G) to guarantee that aspects
of Hs structure align with aspects of Gs structure.
1.2 Flag Algebras
In 2007, Razborov [75] presented a formal model called flag algebras in which a
number of central problems of extremal graph theory can be expressed. Specifically,
this model allows the formalization of several important tools in extremal combi
natorics such as the CauchySchwarz inequality to be applied to problems in graph
theory. In some sense, flag algebras provide a method that enables finding the best
possible way in which to apply the CauchySchwarz inequality to a collection of graph
parameters. This endeavor is completed by way of solving a semidehnite program,
which is convenient given the number of publicly available, optimized, semidehnite
program solvers.
While we will overview the process of solving such a program in the next section,
the remainder of this section will be devoted to providing a conceptual understanding
of hag algebras. Although we will deal primarily with simple graphs in the context
of subgraph densities, hag algebras can also be used to represent oriented graphs
[42, 52, 59, 60, 61, 78] and hypergraphs [9, 10, 43, 70, 77]. For clarity, we will
use pictures of graphs rather than their names as often as possible. For a more
detailed presentation of the theory of hag algebras, we direct the reader to the paper
of Razborov [76], for which he was awarded the Robbins Prize from the American
Mathematical Society in 2013, and the dissertation of Sperfeld [85] from which we
borrow some useful examples.
To dehne hags, we need hrst the notion of a type. If a is the subgraph induced
by all labeled vertices of a graph G, then G has type a and is called a erhag. Note
5
that unlabeled graphs are said to have type 0. Focusing on labeled graphs allows us
to distinguish between isomorphic copies of a graph, an endeavor that is crucial both
in theory and also in computational implementation, where all graphs are labeled.
We will widely view a erflag as its density in an arbitrarily large graph. This
density is formally obtained from probabilities, which we describe here. A collection of
sets Vi,..., Vt is a sunflower with center C if V^flVj = C for every distinct i,j E [t]. For
a collection of uflags Fx,..., Ft, let G be a graph with enough vertices to contain the
sunflower (Fi,..., Ft) with center V(a), that is G(G) > Y^i=i W(F)I G(cr)(i 1).
Choose in V(G) uniformly at random a sunflower (V\,... Vt) with center V(a) and
\Vi\ = y(Fj) for all % E [t]. We denote by p(Fi,..., Ft; G) the probability that
G\vt is isomorphic to Fi for all i E [t\. For two examples of type 0 with a single Fi,
is the probability of picking three pairwise adjacent vertices in G.
For a potentially more insightful example, consider p ^ , v% ; G^j. Fiere,
a = {u} and each Fi has one additional vertex. Thus, we are concerned with the
probability of picking two vertices of V(G) \ {u} with one adjacent to v and the other
Enumerating the possible sunflowers is a key part of flag multiplication.
Before defining flag multiplication, we first describe a flag algerba chain rule that
allows for the derivation of numerous equalities between probabilities. For a graph
G and a collection of uflags Fx,..., Ft, let Â£ < C(G) so that, for s E [t], t is large
enough so that a graph on t vertices could contain the sunflower (ih,... ,FS) with
center V(a) and G has enough vertices to contain the sunflower (Ft, Fs+1,..., Ft) with
center V (a), where Ft is a erflag on t vertices. Then we can rewrite p(F\,... ,Ft;G) as
a sum of the product p(F\,..., Fs; H)p(Fs+1,..., Ft; G) over all erflags Ft on t vertices.
Thinking of this as a statement involving conditional probabilities, we present the
following example regarding the edge density of a graph in which the sunflowers
p ( I ; Gj is the probability of picking two adjacent vertices in G, and p ( V;G
not adjacent to v. There are two possible sunflowers, specifically
6
(Fl,...,Fs) = ( J) and (H,Fs+l,... Ft) = (Ft) where H ranges over all unlabeled
graphs on t = 3 vertices:
p([ ; g) = P ( ] ;V)^(V;G)+^(I ; \/ ) ^ ( \/ ^ g)
+p(I ; T)^(T;g)+^(I ; V)K'*';G)
As the probability of edges appearing in fixed graphs on 3 vertices is easy to calculate,
we can write the edge density of a graph as a linear combination of the probabilities
of graphs on 3 vertices appearing in G as follows:
P
3
3
1
= 3P
V
V ;G
~;G
P
V ;G
p
G
(1.1)
If we abuse notation to interpret the picture of a erflag F as p(F; F') for a
large enough erflag F', then we can state asymptotic versions of notable results. For
example, consider Goodmans 1959 result [47] on the minimum number of triangles
in a graph and its complement.
Theorem 1.3 (Goodman [47]) Let G be
complement contain at least
 m(m 1 )m 2),
< \2m(m l)(4m + 1,)
2 m(m + l)(4m 1),
a graph on n vertices.
for n = 2m,
forn = 4m+l, and
for n = 4m + 3,
Then G and its
triangles.
Asymptotically, this states that G and its complement must contain at least n3+o( 1)
triangles. This can be stated in the language of flag algebras as V + *.* > i
where the addition of flags is the natural addition of real numbers. Mantels Theorem
[69] on ex(n, Kf) can be stated asymptotically as maximizing J subject to V=o
in order to conclude 1 < \.
7
Using the chain rule, we can formally define the multiplication of two erflags F\
and F2. Let l be large enough so that a graph on l vertices could contain the sunflower
(Fi,F2) with center V(a). Then, define the multiplication of flags F\ F2 to be the
sum of the product p(F\, F2, F)p(F; G) over all erflags F on l vertices. To write the
product of two flags as a linear combination, we use all flags (up to isomorphism) that
can be obtained by adding edges to the sunflower (ih, F2) with center V(a) between
V(Fi) \ V(a) and V(F2) \ V(a). For example, the square of the edgedensity can be
expressed using the subgraph densities of graphs on as follows:
where we abuse notation to allow the picture of a graph F[ to represent p(H;G).
Because J is a 0flag on 2 vertices, multiplying it by itself gives no overlap in the
resulting sunflower. Thus, we examine all 0flags on 4 vertices. The weights on each
probability account for the number of ways that a matching appears from a selection
of two vertex pairs.
As another example, consider the following product:
Because ^ and ^ are 1flags on 2 vertices, their product is a 1flag on 3 vertices
containing an oriented P3 with a as its interior vertex. As before, we consider all
the 2 ways to obtain an oriented P3 with interior vertex a when picking vertices to
build the sunflower.
Under this sunflower definition of flag multiplication, we have the ability to write
any quadratic term (of flags) as a linear combination of flags on more vertices. This
process is a critical step in establishing a flag algebra model that can be solved
comput at ionally.
where dotted lines represent the potential of edges to complete the sunflower and
possible ways to complete this sunflower. We weight each term by  to account for
8
As combinatorial arguments frequently use averaging arguments, we describe the
averaging operator, which will enable us to provide a version of the CauchySchwarz
Inequality in the language of flag algebras. Let F be a erflag and G denote F with
no labels. Define qa(F) to be the probability that an induced embedding of a in G
gives a erflag isomorphic to F. The averaging operator for a erflag F, denoted [Fjo,
is defined to be qa(F)G. When a = 0, we omit the subscript and simply write [F].
Viewed roughly, the averaging operator deletes the labels on a flag and multiplies it
by an appropriate probability. With this notation, the CauchySchwarz inequality
can be stated as follows:
Theorem 1.4 (FlagAlgebraic CauchySchwarz Inequality [75]) Letf,g be lin
ear cow,binations of aflags. Then lf2}a \g2\
The inequality lf2}a > 0 is true as an immediate consequence of Theorem 1.4 by
setting g = 1. We turn our attention toward establishing a broad understanding of
the computational methods used in conjunction with flag algebras.
1.3 Convex Optimization
Convex optimization is particularly friendly because any local maximum (or min
imum) is also a global maximum (or minimum). Semidehnite programming is a
subheld of convex optimization that we rely upon for our results. In what follows we
discuss semidehnite programming in a hag algebra context. We direct the reader to
a book by Helmberg [57] for a more thorough treatment of semidehnite programming
for combinatorial optimization.
Our primary goal is to verify an inequality of the form / > d where / is a linear
combination of
bound on / is as good as possible. For a hag algebra statement of Theorem 1.3,
V + corresponds to / and  corresponds to the maximum possible d.
Toward broadly discussing semidehnite programming as used with hag algebras,
let Tf denote the space of all
9
a linear combination of <7flags. As used in linear algebra, let Sn denote the set of
all symmetric, n x n real matrices. In this context, we define a matrix A G Sn to be
positive semidefinite, denoted A y 0, if xTAx > 0 for all x G Equivalently,
A y 0 if every eigenvalue of A is nonnegative.
As a consequence of Theorem 1.4, we have the following corollary for positive
semidefinite A E Sn and x G (A/):
[xtAx]ct > 0.
Thus, if there exists positive semidefinite A G Sn so that
/ d > [xtAx](J,
then we can conclude that / > d. Often times, such as with Mantels Theorem,
additional known or desired flag inequalities (i.e. constraints) are either necessary or
helpful in solving the problem at hand. If C G Rmxra and c G Rmxl so that CA + c
represents a collection of m flag algebra constraints, then our goal is to solve the
following semidefinite program:
maximize d
(SDP) subject to Ox + c > d,
A y 0.
Several examples of this problem formation can be found in [8], [49], [54], [55], and
[58].
There are several benefits to encoding flag algebra problems as semidefinite pro
grams. First, semidefinite programs are closed under duals, which means the dual
program is also semidefinite. More specifically, strong duality of semidefinite programs
guarantees that if a solution to a semidefinite program exists, then the corresponding
dual program has an identical solution. This allows for primaldual methods that
10
alternate solving the primal and dual optimization problems toward the common so
lution to both problems. There are a variety of solvers, including CSDP [20] and
SDPA [45], for semidehnite programs that run very efficiently. Adding to the con
venience of semidehnite program solvers, Vaughan [91] has developed a free software
called Flagmatic with which to encode hag algebras as a semidehnite program.
Although we will not discuss it in our results, we would be remiss not to men
tion the concern of small errors resulting from boating point numerical computations.
Ideally, we would like to simply round decimal entries of the solvers solution ma
trix to rational numbers and maintain a feasible positive semidehnite matrix. This
is generally possible when the eigenvalues of the solution matrix are all sufficiently
large positive numbers. However, if the solution matrix has eigenvalues close to zero,
then rounding entries risks creating negative eigenvalues, in which case we no longer
have a solution. If we assume instead that eigenvalues close to zero are in fact zero
in an optimal solution, then a technique in Babers dissertation [7] is helpful. Un
fortunately, there is no universal solution to finding sharp results using this method.
Many results are found simply by examining constraints and making small changes
to turn a rounded, nonfeasible solution into a feasible solution.
1.4 Obtaining Extremal Examples
In this section, we overview the standard application of hag algebras in determin
ing extremal constructions and provide an example of its use through two proofs of
Mantels Theorem.
The general approach typically begins with a known lower (or upper) bound ob
tained from a construction. Presumably, this bound is the conjectured extremal value.
After translating the extremal problem under consideration into the language of hag
algebras, a semidehnite program is used to obtain an upper (or lower) bound close to
the known bound. As mentioned in the previous section, the resulting solution matrix
is rounded to obtain a feasible solution matrix that is provably positive semidehnite.
11
From this rounded solution, subgraphs that appear with 0 density can be identified.
Using these added structural properties of forbidden subgraphs, stability arguments
are used to obtain the extremal construction.
It is perhaps worth mentioning that, in practice, rounding the solution matrix
solution matrix, the resulting upper (or lower) bound is itself rounded and utilized
for stability arguments in an attempt to determine if it is close enough for those
stability arguments to hold. If stability arguments hold for the rounded bound, then
the process is reversed to determine what bound would be necessary for the stability
arguments to remain true. The solution matrix is then rounded with the goal of
obtaining a bound sufficient for the stability arguments to give the desired result. In
this context, using efficient semidehnite program solvers to obtain precise numerical
bounds is invaluable. Using powerful computational machines allows computations
to be performed on larger flags, which generate more inequalities and contribute to
obtaining precise numerical bounds.
The remainder of this section is dedicated to presenting two proofs of Mantels
Theorem and determining the extremal construction using flag algebras.
Theorem 1.5 (Mantel [69]) A trianglefree graph on n vertices contains at most
^ + o(l) edges.
Proof: A theoretical approach.
Note that (1.1) can be written as
is commonly the last step undertaken. Rather than initially rounding the entire
(1.2)
For a trianglefree graph, we assume
V=o.
(1.3)
12
Combining (1.2) and (1.3) gives
(1.4)
Hv
Since (1 2
2
3
> 0, we have from (1.3) and (1.4) that
0 <
<
1 2 I
1 4 +4 V + 4 V
o(l)
1 4
I+4V
o(l)
T 2 _
1 2 T 
i .1
o(l)
o(l)
:i.5)
<12
I + (1)>
from which we conclude J <  + o(l). Interpreting J as a probability, we have that
the number of edges in a trianglefree graph is at most (") = yf + o(l), as desired.
Toward determining the extremal construction for Mantels Theorem, assume
V = 0 and J = Under these assumptions, (1.5) gives
0 < 1 2 
1
+ o(D = 3
o(l).
Since > 0 by its nature as a probability, the above inequality implies =0.
Thus V and are the only unlabeled flags on 3 vertices that appear in the
extremal construction. It is straightforward to argue that constructing an extremal
graph with this restriction on 3vertex subgraphs gives a balanced, bipartite graph.
Proof: A computational approach.
Note from the previous approach that it suffices to show J < \ + o(l). Our
intent is to find ci, C2, C3 G R so that
0 < ci
c2
C3
V
13
for every graph. After summing this inequality with (1.4), we have
I
< Ci
1
h C2
3
 + c3 ) \/ +o(l).
:i.6)
Notice that for our desired result we need ci < c2 < and c3 <
Let A = (cb) be a positive semidehnite matrix over ]R. Then for x
^ vm J ),we have xAxT > 0. Thus,
0 <
i) (a cV i
0# ? V% f I 7 I \ V* V*
J \c b I \
T
= a
a + 2c
3
a + 2c
6( V+ V
b + 2c
3
b + 2c
2c
/ *7
+6^^ +o(l)
3 V +o(l).
Setting ci = a, c2 = and c3 = then (1.6) implies
T 1 + a + 2c 2 + b + 2c ^ p m
1 < +^ +3V + (1)
< max < a, 
max < a
3 3
1 + ci + 2c 2 + b + 2c
V ) +(!)
"5
o(l),
3 3^
where the last line follows from 1 = + + V since V
represent the above inequality as the semidehnite program
Minimize d
(1.7)
0. We can
subject to a < d,
(.SDP) <
1 \a\2c j
3 ^
2 ~fr~l~2c ^ j
3 a
( & ) A 0.
Solving (SDP) with a computational solver gives a solution of (_^55 T55) Thus, (1.7)
implies J < max {, , + o(l) =  + o(l), as desired.
14
Toward determining the extremal construction, assume J = The solution of
(SDP) and (1.6) give the following inequalities:
1
2
1 <
1
6
1
3
2 V + C).
V +(1)
Since V = 0, we have !=. + .+ V Subtracting this from the above
inequality gives
0 < + (1)>
which implies # = 0, as before. Determining the extremal construction finishes
as before.
15
CHAPTER 2
DENSITY BOUNDS IN MULTIPARTITE GRAPHS
In this chapter we are interested primarily in multipartite graphs. A graph is
kpartite if its vertices can be partitioned into k independent sets, which are referred
to as parts. A fcpartite graph is complete if there is an edge between every pair of
vertices from distinct parts. Turan graphs are special cases of complete multipartite
graphs in which the number of vertices in each part differs by at most one.
Considering the existence criteria for triangles in graphs established in Mantels
Theorem [69] and the work toward computationally identifying triangles in graphs
[4, 26, 62], it is perhaps natural to pursue other existence criteria for triangles in
specific graph classes. The goal of this chapter is to use the theory of flag algebras to
identify a condition that guarantees the existence of a triangle in 4partite graphs.
2.1 Triangles in Multipartite Graphs
While Mantels Theorem and Turans Theorem are typically introduced as count
ing thresholds, they can also be viewed as edgedensity thresholds. For example,
Mantels Theorem can be stated in the following form:
Theorem 2.1 (Mantel [69]) Every bipartite graph G with \E(G)\ > con~
tains a K3.
In a bipartite graph, all of the edge density occurs between the two parts of the
partition. In multipartite graphs with more than two parts, it is natural to wonder if
there is an edgedensity threshold between parts that also guarantees a if3 or, more
generally, a Kfor some fceN.
To this end, let G be an Gpartite graph with partition V\, V2,..., Ve. For brevity,
define G := U(G) and G := if(G). For i yt define dij to be the edge density
between U and Vj, that is
d
G[Vi U Vj\
m m
16
Let d\ denote the minimum density between parts in G that guarantees that G
contains a Kk, that is
d* := min{c G K: mindy > c => Kk C G}.
i,j
Motivated by a question of Erdos [35], in [18] Bondy, Shen, Thomasse, and
Thomassen determined h3 = r, the golden ratio. They also showed that ex
ists and equals  and that d\ > 0.51. They speculated that d\ >  for all hnite l.
However, Pfender [72] proved d\ =  for Â£ > 12. Pfender also proved that d\ = 5y
for large enough i.
Our work determines an upper bound for d\. The bulk of this effort lies in formu
lating constraints so that flag algebra computations are implemented on a balanced,
4partite graph.
2.2 A Proposed Extremal Construction
Generalizing a lemma from Bondy, Shen, Thomasse, and Thomassen [18], Sperfeld
[84] provides a construction that lower bounds d\. This construction is a blow up of
a 4partite graph with at most 3 vertices in each part. As we have not specifically
mentioned it previously, the blowup of a graph G is obtained by replacing each vertex
with a set of independent vertices and adding edges between vertices if they are in
sets that correspond to adjacent vertices in G. The general structure of Sperfelds
construction is shown in Figure 2.1. For the skeptical reader, we describe the graphs
Figure 2.1: The general structure of a conjectured extremal graph for d\.
underlying structure and include a proof that it is id3free, which implies that any
blowup is also id3 free.
17
Proposition 2.2 Let G be a graph with vertex set Xi,yi,Zi for i E Z4 such that
G[xq,xi,X2,xz\ = C4, G[xi,yi, Zi,yi+2, Zi+f\ = fd3)2 with bipartition denoted by indices,
and yoyi,y2V3, Z0Z1, Z2Z3 E E(G). We have that G is Ksfree.
Proof: Since each {xi,yi,Zi: i E Z4} is an independent set, any triangle must
witness distinct indices. Since iV(ay) = {ay_i, ay+4, y%+2, Zi+2}, if a triangle contains
Xi, then it must also contain ay_ 1 or ay+4. However, this is not possible because
N(xi) fl N(xi+i) = 0. Thus, a triangle does not contain any ay.
Since G[y0,..., y3\ = G[z0,..., z3\ = 2K2, a triangle must contain at least one ty
and at least one Zj. By symmetry, it suffices to consider if a triangle can contain y0 and
yi This is not possible because N(y0) = {yi,x2,y2, z2} and N(yi) = {y0, x3, y3, z3}.
Therefore, G is fd3free.
Using heuristics to optimize the sizes of each set in the blowup extremal con
struction, we get the following lower bound for d\\
d\ > 0.511342. (2.1)
For the interested reader, we consider a vertical symmetry in Sperfelds construction
[84] that he proves optimizes the lower bound given by the general structure presented
in Figure 2.1. We view ay, ty, for i E Z4 as independent sets of vertices and assume
ar0 = lari I, y0 = M, \z0\ = N, ar2 = ar3, y2 = y3, and z2 = N and then use
Mathematica to numerically maximize the minimum of dij for i yt j e Z4. Specihcally,
this is the minimum of the set
{ax + zc, x2 + y2, a2 + b2,1 ax}
when viewing x,y,z as the respective proportions of Xi,yi,Zi in ay + ty + {zf for
i E {0,1} and a,b,c as the respective proportions of Xi,yi,Zi in ay + \yf\ + {zf for
{2,3}.
18
2.3 A Flag Algebra Model
In this section, we consider a first approach to modeling a id3free 4partite graph
using flag algebras. Since a 4partite graph is frequently viewed as properly 4vertex
colored, it is natural to consider labeling vertices to specify partite sets. For instance,
labeling vertices of flags from Z4 can be viewed as specifying in which partite set a
vertex resides.
In this framework, establishing constraints for a flag algebra model becomes very
straightforward. For instance, ensuring nonedges between vertices of the same label
do not occur is easily accomplished by setting each edge density between vertices of
the same label to be 0. To ensure that triangles do not appear, we would also set
the density of the remaining 4 possible proper vertex labelings from Z4 to id3 equal
to 0. In conjunction with the previous 4 constraints on edge densities, this suffices
to guarantee our flag algebra constraints model a 4partite, id3free graph. These
constraints are pictured in Figure 2.2.
0123 12132323
IIIIVVVV
0 1 2 3 0 0 0 1
Figure 2.2: Flag algebra constraints for vertexlabeled graph modeling a 4partite,
K3free graph.
While adding constraints to the semidehnite program often improves the solvers
computation time, the size of the semidehnite program is the primary factor impacting
computation time. With the size of the semidehnite program dependent upon the size
and type of hags used, the number of hags on a hxed number of vertices is a reasonable
indicator of the semidehnite programs size. In large part this is because variables for
the semidehnite program are generated by multiplying pairs of hags, as seen in the
computational proof of Theorem 1.5.
Considering that there are 16 hags on 2 vertices, 72 hags on 3 vertices, and 407
19
# Vertices 2 3 4 5
Vertex4Color Model 16 72 407 2235
Edge3Color Model 3 8 28 137
Figure 2.3: Comparing the number of flags on small vertex sets.
flags on 4 vertices satisfying the constraints in Figure 2.2, the number of flags grows
very quickly as their size increases. Thus, computations for this vertexlabeled model
are restricted to using very small flags that give weaker bounds. Figure 2.3 compares
the number of flags on small numbers of vertices in this model with those for the
model presented in the next section. The computational limitations resulting from
the large number of small flags in the model presented in this section serve as our
primary motivation for using an alternative model.
2.4 Multipartite Result
We proceed under the assumption that Sperfelds construction is extremal for
d\. We begin by describing some additional assumptions we can make. Then, we
discuss how flag algebras can be used to encode both the known structure of and
our assumptions about an extremal graph. Finally, we use our model to obtain
computational bounds for d\.
Instead of coloring vertices to denote partite sets, we seek to partition vertices by
3edgecoloring a complete graph under specific constraints. Specifically, we seek to
encode whether vertices are in the same partite set using red, blue, and green edges.
To this end, we will view red edges as corresponding to edges in a 4partite graph,
blue edges as corresponding to nonedges between vertices in distinct partite sets of
a 4partite graph, and green edges as corresponding to nonedges between vertices in
the same partite set of a 4partite graph.
A solid edge, J will represent a red edge in the 3edgecolored complete graph.
A dotted edge, j will represent a blue edge in the 3edgecolored complete graph.
20
A dashed edge, ^ will represent a green edge in the 3edgecolored complete graph.
A multiedge will represent any of the corresponding colored edges. For example,
^ denotes the possibility of either a red or blue edge in the 3edgecolored complete
graph.
We claim that we can assume our graph is balanced in two ways. First, we can
assume that the partite sets are balanced; otherwise we can blowup every part by
an appropriate constant to balance the sets. Note that this scaling of entire parts
preserves the densities between parts. Also, we can assume that edge density between
parts is balanced, that is dij = dki for all 1 < i < j < 4 and 1 < k < Â£ < 4, because
edges can be deleted between parts of an unbalanced extremal example to obtain an
extremal example in which every d^ witnesses d\.
With the 3edgecolored model in mind, we wish to formulate flag algebra con
straints under which a complete 3edgecolored graph represents a balanced, 4partite,
Jd3free graph. Figure 2.4 summarizes the sufficient flag algebra constraints discussed
in Lemma 2.3. Although the last constraint in Figure 2.4 and Lemma 2.3 appear
different, we show within the proof of Lemma 2.3 that they are equivalent.
Constraint Meaning in Complete Graph Model Forced Graph Property
V= no red Jd3 no Ks
V = transitivity of green edges multipartite
=0 no K5 in red and blue 4partite
\ i 4  of edges are green balanced partite sets
3* .4 3. .4 2 m = m r z r *2 equal density of weighted families balanced d^
Figure 2.4: Flag algebra constraints for 3edgecolored complete graph modeling a
balanced, 4partite, Jd3free graph.
The evident difference in the number of flags satisfying the constraints under both
21
models, as presented in Figure 2.3, motivates our choice to model this problem as a
3edgecolored complete graph.
Lemma 2.3 Let G be a 3edgecolored complete graph using colors red, blue, and
green. Let H be a graph obtained from, G by deleting blue and green edges. If G
satisfies
then H is a Kafree, balanced, fpartite graph with dij = du for all 1 < i < j < 4 and
1 < k < Â£ < 4.
Proof: Let G be a 3edgecolored complete graph on n vertices satisfying the given
indicates that the probability of three vertices in G spanning a red triangle is 0. Since
red edges in G correspond to edges in H and there are no red triangles in G, H is
Ksfree.
Since dashed edges in flags correspond to green edges in G, = 0 indicates
that if edges xy and yz are both green in G then the probability of xz being red or
blue is 0. Hence, xz must be green. This transitivity implies that the green edges of
G induce a disjoint union of complete graphs, say G'. Furthermore, = 0 implies
that every set of 5 vertices in G must induce a green edge. Thus, there can be at most
4 components in G'. Since each component of G' corresponds to an independent set
in H, H is 4partite.
Toward a balanced partition, for i e Z4, let ay be the fraction of vertices in each
part of H. That is, ay > 0 for i E Z4 and ay = 1. Notice that the proportion
of green edges in G satisfies the following:
0
0
flag equalities. Since black edges in flags correspond to red edges in G, V =o
22
By the CauchySchwarz Inequality we have
as n > oo. Thus,
: >i.
* 4
The added constraint of j = \ forces equality in the CauchySchwarz Inequality,
which occurs precisely when ay = ay for i, j E Z4. Hence, the parts of H are balanced.
It remains to show that edge densities between partite sets in H are balanced.
To this end, let i,j, k,Â£ E Z4 such that i ^ j and k 4 l. Observe that dijdke + o(l),
where o(l) as n > oo, is the probability of independently picking two pairs of vertices
(vi, Vj) G (Vi, Vj) and (vk, ve) E (14,14) so that v(Vj and VkVg are red edges. For distinct
i,j, k,Â£ E Z4, the products dfJ + o( 1), dijdik + o( 1), and dijdke + o( 1) can be represented
as the respective conditional probabilities
Thus, if the conditional probabilities in (2.2) are equal, then
dij + o(l) = dijdik + o(l) = d^dke + o(l).
Under this assumption, d^ = dke as n > oo for all i ^ j E Z4 and k ^ Â£ E
Z4, which gives the desired balance. In order to implement this sufficient equality
within flag algebra computations with the other constraints, we desire an unlabeled
representation of this equality. In what follows, we examine how to establish d^ +
o(l) = d^dik + o(l), which is sufficient to ensure d^ = dke for all i ^ j E Z4 and
k 4 Â£ E Z4.
We work backwards from the desired equality and assume
d^j + o(l) = d^dik + o(l). (23)
23
First, we examine hf + o(l) = P
. Notice that there are Q ways
to pick i yt j e Z4 with 2 subsequent choices from which part to pick each pair. Then,
there are (^)2 + o(l) ways to pick two vertices from each part. Thus,
4\ /n\4 . 3n4 . ,
2 Mi) + 0(1) = ig + 0
f 2.43
Similarly, for ckjdik + o(l) = P
,2
there are 4! ways to pick distinct
i,j, k G Z4, + o(l) ways to pick 2 vertices in Vi, and  vertices to pick in both
Vj and Vfc. Thus,
SKT34 /n \3 fin4
2.5
3 4
r *2
4!'i)' = ^^4
Combining (2.4) and (2.5) in (2.3) gives the following equalities:
d2ij + o(l) = dijdik + o(l),
We use the averaging operator to represent the labeled flags in (2.6) as linear combi
nations of unlabeled flags.
Listing all of the possibilities for the left hand side of (2.6) gives
3* .4
l9" z
(2.7)
For the averaging of each labeled flag, we count how many of the 4! vertex labelings of
an unlabeled flag result in the corresponding labeled flag. Since all vertices must be
incident to one green(dotted) and two red(solid) edges in M there are 4 choices for
label 1 that all force the choice for label 2, which in turn gives 2 ways to appropriately
label 3 and 4. Since 1 must be incident to all three edge colors in both of the 2
24
choices for label 1 force the choice of the remaining labels. Since 2 must be incident
to all three edge colors in both of the 2 choices for label 2 force the choice of the
remaining labels. Since all vertices must be incident to all three edge colors in ,
all 4 choices for label 1 force the choice of the remaining labels. Therefore, from (2.7)
we have
4 2 * 4 k 4 r y
W
r *2
m :.^l :K 31
4! 4! 4!
Listing all of the possibilities for the right hand side of (2.6) gives
2.8
r 2
3_ _4
Since
3 _ 4
r *2
3.__.4 3m__m4
, and
r 2
3_ _4
IT 2
3_ _4
:r 2
3. .4
(2.9)
M + H
]/j\ each contain a V the first constraint gives
0. Since there is only one green (dashed) edge in [Xj there
are 2 ways to label 1 and 2, both of which force the labels of 3 and 4. Since there is
only one green(dashed) edge in there are 2 ways to label 1 and 2, after which
there are 2 ways to label 3 and 4. Since 1 must be incident to all three edge colors
in JXJ there is only 1 choice for label 1, which forces the remaining labels. Since 2
is incident to all three edge colors in there is only 1 choice for label 2, which
forces the remaining labels. Since there is only one green(dashed) edge in JX] there
are 2 ways to label 1 and 2, both of which force the labels of 3 and 4. Therefore, from
(2.9) we have
2
4!
2 2
XT
1
4!
1
4!
RlXEil
12.10)
Simplifying the combination of (2.6), (2.8), and (2.10) gives
H fxj I 1 Jxl +4 J>:J
+ 2
M + KI
which is the desired unlabeled constraint that forces the desired equality in (2.3)
25
As a result of this Lemma, we have constraints in the language of flag algebras
whose addition to flag algebra computations give bounds for the desired graph. Using
these constraints, we can run computations to find an upper bound for d\.
Theorem 2.4
d\ < 0.513243.
Proof: This follows from a standard application of the plain flag algebra method.
We maximize J with the constraints listed in Lemma 2.3. The computations were
performed on 7 vertices and a solution to the semidehnite program was found using
CSDP.
For comparison with the previously mentioned lower bound (2.1), we have the
following regarding d\\
0.511342
The observant reader may have noticed that this 3edgecolored model does not
fully model Sperfelds [84] construction, which is pictured in Figure 2.1. In particular,
the 3edgecolored model does not enforce the refined partition into 12 parts with the
appropriate edges between parts. Below, we discuss a way in which to add an edge
color, say orange, to refine the 3edgecolor model toward Sperfelds extremal example.
In this 4edgecolored model, red and blue edges retain their same meaning, and
green and orange together provide the 3part structure to each partite set. Specifically,
orange edges are used to refine each part in the 4partition by recoloring green edges
to indicate when vertices in the same part are also in the same part of the refinement.
A dashdotted edge, J will represent an orange edge in the 4edgecolored complete
graph.
The constraints for this 4edgecolor model use many of the former constraints.
Figure 2.5 presents constraints that force the refinement of a 4partite graph into
a 12partite graph. However, these constraints neither give the adjacency structure
26
Constraint Meaning in Complete Graph Model Forced Graph Property
V=o no red K% no Ks
V = neartransitivity of green edges multipartite in red/blue
^ = no K5 in red and blue 4partite in red/blue
T i + i 1 4  of edges are green/orange balanced 4partite in red/blue
V =0 transitivity of orange edges multipartite in red/blue/green
'J: 5=0 no K4 in green 3partite in green
Figure 2.5: Flag algebra constraints for 4edgecolored complete graph modeling a
12partite, iGfree graph.
pictured in Figure 2.1 nor give the balanced nature of Sperfelds [84] optimal con
struction. Implementing this 4edgecolor model did not obtain better results than
the 3edgecolor model. For the motivated reader, we note that it may be possible
to improve our bound using the 4edgecolor model by adding constraints to bal
ance the graph; we did not exert much effort toward establishing these flag algebra
constraints.
27
CHAPTER 3
INDUCED CYCLES
In this chapter, we consider an inducibility problem for cycles in which we will
maximize the number of induced cycles in a graph of order n. For a graph H on k
vertices, let denote the maximum number of induced copies of H in an nvertex
graph. Similar to Proposition 1.1, we have the following proposition regarding
Proposition 3.1 For any kvertex graph H and any n> k,
7vH(n+ 1) 7rH(n)
m o '
Proof: Let G be a graph on n + 1 vertices with 7177(71 + 1) induced copies of H.
Since each copy of H in G has k vertices and G has n + 1 vertices, G must have a
vertex v in at most k7TH^1"1 copies of H. In G v, there are at least
/ , n KH(n+ 1) nH(n+l)
TTnyTi, + 1) k = (n + 1 k)
n+ 1
n + 1
copies of H. However, the n vertex graph G v has at most copies of H. Thus,
. ,. 7inin + 1)
(n + 1 k) < 7Tu(n).
Using the combinatorial equality
n+1
k
n+1
n + 1 f n'
n+1 k\k}'
we have
7rH{n + 1) / 7TH(n)
"V) C)
as desired.
The inducibility of H is defined to be
i(H) := lim lr"(")
28
Since is nonincreasing by Proposition 3.1 and is bounded below by 0, this limit
exists.
The balanced iterated blowup of a graph H is a blowup of H where every blowup
set both differs in size by at most one vertex and also induces a balanced iterated
blowup of H. An example of the iterated blowup of C5 is provided in Figure 3.1.
Figure 3.1: Progression of obtaining a balanced, iterated blowup of C4.
In 1975, Pippenger and Golumbic [74] noted that this balanced iterated
up for any graph H on k vertices gives a general lower bound of i(H) >
Given this general lower bound for inducibility, a natural endeavor is to determine
for which graphs H the inducibility is witnessed by the balanced iterated blowup of
H. Equivalently, we may ask for which graphs H the balanced iterated blowup of
H is an extremal graph for i(H). In a random graph setting, Fox, Huang, and Lee
[44] recently announced that the inducibility of almost all graphs is witnessed by its
balanced, iterated blowup. However, this is certainly not always the case. Answering
a question of Bollobas, Egawa, Harris, and Jin from [16], in [53] Hatami, Hirst, and
Norine show that the inducibility of sufficiently large balanced blowups is attained
by a blowup rather than an iterated blowup. Further, the collective examination of
the inducibility of small graphs [16, 17, 24, 41, 58] also shows this is not the case for
all graphs on at most 4 vertices except complete and empty graphs. A nice overview
of the inducibility of small graphs appears in [40].
In their 1975 paper, Pippenger and Golumbic made the following conjecture:
Conjecture 3.2 (Pippenger, Golumbic [74]) For k > 5, i(Ck) is witnessed by
blow
fc!
kkk
29
the balanced iterated blowup of Cf.
For a long time it was believed that progress on this conjecture was intractable even
for C5. When Razborovs theory of flag algebras was introduced over 30 years after
the conjecture, many thought this powerful new tool could be used to finally make
progress on the conjecture. While this is true, much like the multipartite problem
in the last chapter, the straightforward application of Cauchy Schwarz using flag
algebras under the standard approach do not give upper bounds sufficiently close to
the conjectured lower bound for stability arguments to establish exact results.
In the case of C5, the iterative nature of the extremal construction is what causes
problems for the standard flag algebra approach. While there are many graphs whose
conjectured extremal construction is an iterated blowup [73], we are not aware of
many applications of flag algebras which completely determine an iterative structure.
FalgasRavry and Vaughan [42] proved an iterative extremal structure for small ori
ented stars that was generalized to all stars by Huang [61], though he used different
methods. Recently, Hladky, Krai, and Norine [60] announced a result on oriented
paths of length 2.
In the next section, we will discuss the paper of Balogh, Hu, Lidicky, and Pfender
[11] confirming Conjecture 3.2 for k = 5 for graphs of order 5m for some m G N.
Beyond their submitted paper with Volec and Young [12] using a similar approach,
we are unaware of any further iterative extremal structures proved using flag algebras.
Their general approach will be used in our results in Section 3.2.
3.1 Induced 5Cycles
As mentioned previously, Conjecture 3.2 was confirmed for k = 5 on graphs with
the appropriate number of vertices by Balogh, Hu, Lidicky, and Pfender [11], The
method used is slightly different from the standard approach, which we briefly review
here. In the standard approach, flag algebra computations provide an upper bound for
i(C
30
the solution to the semidehnite program established from flag algebra inequalities,
we identify subgraphs that appear with very low probability and then use this fact
within stability arguments to obtain an extremal construction.
The specific shortcoming of the standard approach in application to short in
duced cycles arises in the upper bound obtained from flag algebras. Specifically, the
lower bound for 2(65) obtained from the iterative extremal construction and the cor
responding upper bound obtained using flag algebra computations with flags on 7
vertices are as follows:
0.03846153... = < i(C5) < 0.03846157
26
These bounds leave a gap of approximately 4 x 108. While this gap seems very small
it is still too large for the standard approach to overcome given the iterative structure
of the extremal construction. In large part, this is caused by small numerical errors
having a large relative effect on the density of subgraphs appearing with 0 density.
These small numerical errors have less relative effect on the density of subgraphs
appearing with high density. The iterative method capitalizes on this lesser impact by
using such subgraphs instead. The prevalence of these high density subgraphs provide
information about the general top level structure of the extremal construction. For
induced C5S, the high density subgraphs guarantee enough edges between parts so that
the extremal graph looks like the balanced blowup of C5 with extra edges possible
in each part.
For the iterated approach, a collection of high density subgraphs are identified
using flag algebra computations. The high density subgraphs used are specifically the
family of all 7vertex graphs obtained from a C5 by either duplicating two vertices
once each, or duplicating a single vertex twice. Stability arguments are used to show
that this top level structure is in fact a balanced blowup of C5 and then induction is
used to confirm the iterated structure of the extremal construction. For our result in
the next section, we use the family of all 8vertex graphs obtained from a Cq in the
31
similar way.
3.2 Induced 6Cycles
We begin by defining some notation. Let CgX denote the (k l)times iterated
blowup of Cq. In this notation, C\x is simply Cq. Let Qn be the set of all graphs
on n vertices, and let C&{G) denote the number of induced copies of C$ in G. Define
Ce(n) = maxC6(G). A graph G E Q is said to be extremal if Ce(G) = Ce(n). If n is
a power of m, we can exactly determine the unique extremal graph and thus Ce(n).
Theorem 3.3 For k > 1, the unique extremal graph in Qok is CgX.
Figure 3.2: The extremal graph for i(Ce), CgX.
To prove the iterative structure of Cg x in Theorem 3.3, we first prove the following
theorem about the top level structure of Cg x.
Theorem 3.4 There exists no such that for every n > no,
5 5
Coin) =
i=0 i=0
where Y^,=o xi = n and xo,xi,... ,x$ are as equal as possible.
Moreover, if G E Qn is an extremal graph, then V(G) can be partitioned into sets
X0, Xi,..., X5 of sizes xo, x\,..., Â£5 respectively, such that for 0 < i < j < 5 and
Vi E Xi; Vj E Xj, we have vpVj E E(G) if and only if j i E {1,5}.
The proof proceeds similarly to the proof in [11] with the proof of Theorem 3.4
appearing in Section 3.2.1 and the proof of Theorem 3.3 in Section 3.2.2. For the
interested reader, certificates associated with flag algebra computations will be made
available with the published results.
32
3.2.1 Top Level Structure of the Extremal Graph
We consider densities of 8vertex subgraphs. Based on the conjectured extremal
graph, we consider two families of graphs that are rather prevalent. In particular, let
C22 be the family of graphs that can be obtained from Ce by duplicating two vertices
and C3 be the family of graphs that can be obtained from Ce by duplicating a single
vertex twice. For both families, the presence or absence of edges between the original
vertices and their duplicates specify graphs in the family, see Figure 3.3.
Figure 3.3: Representative graphs for C22 and C3 where dotted edges denote the
possibility of edges. The family C22 consists of 9 graphs and the family C3 consists
of 4 graphs.
We let C22 represent any graph in C22 and C3 represent any graph in C3. For
convenience, we will also use C22 and C3 to denote the densities of these respective
graphs, that is the probability that randomly selecting 8 vertices induces the cor
responding 8vertex expansion of Cq. Additionally, for a set of vertices Z, C22(Z)
and C3(Z) denote the respective density of C22 and C3 containing Z. That is, for a
graph G on n vertices C22(Z) and C3(Z) are the respective number of C22 and C3
containing Z divided by (gljfj)
Note that for CqX in the limit as k > oo we have 4C22 15C3 = 0. Similar
to [11], we need a positive lower bound for our approach to work. In what follows,
a = 4.95. The limit values of interest in the balanced iterated blowup CqX where
33
k > oo, are
o
Q rj
C22C3
4
24
1555 '
5040
55987
0.0154340836, and
3a
T
1344
55987
0.00090020897.
We obtain comparable values using flag algebra computations.
Proposition 3.5 There exists n0 such that every extremal graph G on at least n0
vertices satisfies
O <
5403179531366257035327
< 0.015437656, and
350000000000000000000000
^ 335999533394428927782939
C22C3 > > 0.0008999883348.
4 40000000000000000000000000
Proof: We apply standard flag algebra computations. The first inequality is ob
tained directly with computations on seven vertices. The second inequality is obtained
by minimizing C22 on eight vertices with the added constraint that Ce > y^.
It is worth mentioning that flag algebra bounds from computing on 8 vertices
obtain better bounds on Ce. However, these bounds are not sufficient for the standard
flag algebra approach to be successful.
Let G be an extremal graph on n vertices with n > no from Proposition 3.5. Let
Z be the family of all induced C6s. Let Z = z0z\ z5 be an induced C6 maximizing
(C22(Z) a C3(Z)) ('" 6) > ^ (C22(y) a C3(Y)) in 6
>
\z\
1 1 YEZ
(4 C22 3a C3) g
4 C22 3a C3 fn 6
280 1 2
With Proposition 3.5 and a = 4.95, we have
C22(Z) a C3(Z) >
4 0.0008999883348
> 0.008328321483.
28 0.015437656
(3.1)
34
For the remainder of this section, unless otherwise specified, indices and arith
metic thereon are in Zq. We define sets of vertices Zi that have the same neighbors
in Z as Zi. Formally,
= {veV(G):G[(Z\z1)Uv]^C6}.
Notice that Zi n Zj = $ for i yt j. We call a pair ViVj funky if v(Vj is an edge while
ZiZj is not an edge or vice versa, with i ^ j, Vi E Zi and vj E Zj. Notice that
G[Z U {vi,Vj}\ ^ C22, which means every funky pair destroys a potential copy of
C22(Z). Let Ef denote the set of funky pairs. Thus, (3.1) implies
z I2
Y\zi\\zj\ \Ef \ 4.95 Y^ 2^~ > 0.008328321483
i
For any choice of disjoint sets X* C V(G), let T := V(G) \ (J Xj. To reflect the
more general choice of sets we adjust our definition of a funky pair to a pair Vi,Vj
with Vi E Xi,Vj E Xj where v(Vj is an edge when % j {1,5} or when v(Vj is not
an edge when i j E {1,5}. In the following definition we slightly abuse notation
by using Ef to reflect our adjusted definition of funky pairs. For normalization, let
/ = xi = ~\ Xi, and t = ^\T\. Choose sets Xj C Zi (possibly Xj = Zi) such that
the left hand side of
2 XiXj 2/ 4.95 Y X1 > 0.008328321483 (3.2)
i
is maximized.
Claim 3.6 The following inequalities are satisfied:
0.16582 < Xi < 0.16751, (3.3)
/ < 0.000003, and (3.4)
t < 0.00031. (3.5)
Proof: We must solve four quadratic programs. The objectives are to minimize x0,
maximize xo, maximize /, and maximize t, respectively. The constraints are (3.2),
35
n 6
2
t + xi = 1) and x%, f,t > 0 in each case. By symmetry the bounds for Xo hold for
all Xi.
Here we describe obtaining the lower bound on xo in (3.3). We need to solve the
following program (P):
(P)
minimize Xq
subject to t + J2ixi = 1>
2 Y,i 0.008328321483,
Xi, f,t> 0.
We claim that if (P) has a feasible solution S then there exists a feasible solution S' of
(P) where S'{xo) = S(xo), S'(t) = S(t), S'(f) = 0, and S'{xi) = (l S(xo)) for i E
{1,..., 5}. Since each ay with i E {1,..., 5} does not appear in the objective, we only
need to check if (3.2) is satisfied. The left hand side of (3.2) can be rewritten as
5
2xq Xi + 2 XiXj 4.95 xj 2/
l
5
= 2ai0 Xi (xj Xj)2 0.95 a^ 4.95aiQ 2/.
l
Note that Ei
minimized when / = 0. The term Yli=ixh subject to Ei=i Xi being the constant
1 xo t, is also minimized when ay = Xj for i,j E {1,...,5}. Hence, (3.2) is
satisfied by S' and we can add the aforementioned constraints to bound xq. The
resulting program (P') is
minimize Xq
(P')
subject to t + aio + 5y = 1,
<
10aioy 0.95 5y2
4.95a?d > 0.008328321483,
xo,y,t > 0.
36
We solve (P') using Lagrange multipliers. This work is delegated to Sage [30] and the
Sage script is provided as a supplemental hie named solvexi. sage. Since finding
an upper bound for ay occurs on the same constraints, the same script can be used
simply by changing the objective to maximization.
We turn our attention to finding the upper bound for / in (3.4). Since / is only
constrained by (3.2), we can maximize / by maximizing
2 XiXj
i
4.95J^ay2 0.00832831837
i
= J2(xi Xjf 0.95 J^ay2 0.008328321483,
i
which occurs when ay = Xj, as before. Substituting ay = pp gives
(P")
maximize
subject to
/
1 > t > 0,
03 (Â¥)2
2/ > 0.008328321483,
/> o.
We solve (P") using Lagrange multipliers by making the appropriate changes to the
Sage code.
We proceed similarly for an upper bound on t. Since t is only constrained by
t + JO ay = 1, the left hand side of (3.2) is maximized when / = 0 and ay = Xj, as
before. Substituting ay = ^ gives
maximize t
(p/")
subject to 1 > t > 0,
0.3 (^)2 > 0.008328321483.
Making the appropriate changes to the Sage code enables solving (Pm) using Lagrange
multipliers.
37
For any vertex v E Xi, we use df(v) to denote the number of funky pairs incident
to v after normalizing by n, that is the funky degree of v. If we move v E Xi to T,
then the left hand side of (3.2) will decrease by
 2 Xj 2df(v) 2 4.95 ay + o(l)
U V &i
If this quantity was negative, then the left hand side of (3.2) could be increased by
moving v to T, which would contradict our choice of Xi. Together with (3.3), this
implies
df(v) <^2xj 4.95 Xi + o(l) < 1 5.95 ay + o(l) < 0.013371, (3.6)
JX
which we will use to obtain a contradiction in Claim 3.9. For use in obtaining this
contradiction, we first include a brief argument regarding the number of funky edges
in an induced Cq.
Claim 3.7 Any induced Ce in (J Xi using a funky pair must use at least two funky
pairs.
Proof: Suppose for contradiction that C = vqV\ v$ is an induced C$ using
exactly one funky edge. Without loss of generality, let v0 be incident to this funky
edge. Notice that C v0 is an induced P5. Since there are no funky edges in C v0,
C Vo must have either at most one vertex in each Xi or every vertex in a single Xi.
Assume C v0 has at most one vertex in each part. Since C uses a funky edge, v0
must be in the same part as another ry for i E {1,..., 5}. If v0 is in the same part as
some Vi for i E {1, 2, 3}, then vqV5 and VoVi+\ are both funky edges, a contradiction.
Similarly, if no is in the same part as some ?y for i E {4,5}, then vqV\ and ?y_i are
both funky edges, a contradiction.
Thus, every vertex of C no is in the same part, say Xi. Since C uses a funky
edge, no can not be in Xi. If no G A}_i U A}+i, then U0U2, U0U3, U0U4 are funky edges
38
used by C, a contradiction. If vq ^ Xj_i U Xi+i, then vqVi vqv^ are funky edges used
by C, a contradiction. Therefore, C can not use exactly one funky edge.
For brevity, we let df denote max{d/(u): v G UX*}. Similarly, we let xmax and
Xmin denote max{ay: i G Z6} and min{ay: i G Z6}, respectively. For clarity in
showing there are no funky pairs, we first present a particularly detailed case. Should
the trusting reader wish to skip the details, the proof of Theorem 3.4 continues on
page 44.
Claim 3.8 There are at most tn 10(xmaxn)3 induced Ces containing both exactly one
vertex from, T and a funky pair uv.
Proof: Considering that a Ce without its vertex from T is an induced P5 using
exactly one funky pair, we examine the ways to embed such a P5. We argue that
there are no more than t lOx^^n4 such Cqs in three cases based on the type of funky
pair used.
The following limitation on choosing additional vertices in building a P5 applies
to all three cases, so we include it here. If X* has a vertex and we pick 2 additional
vertices from Xi+i to use in the induced P5, then we have an induced P3 whose
endpoints have the same adjacency pattern to vertices outside of Xi+i. However,
picking a third additional vertex from Xi+1 forces the vertex in Xj to have 3 neighbors
and picking a third additional vertex from Xj U Xi+2 creates a C4. Since a similar
argument holds if we picked 2 additional vertices from Xj_1; we can pick at most 1
additional vertex in each Xj.
Case 1: u G Xj, v G Xj with i j G {1,5}.
Without loss of generality, let u G X0 and v G X\. We begin by limiting where
and how many vertices can be chosen from XjS.
Because an induced P5 with uv as its only funky pair can not have u and v as
endpoints, we must use at least one additional vertex in X0 U X\. However, using
39
an additional vertex from both X0 and X\ creates an induced P4 (or a graph with
a cycle). Assuming this choice of additional vertices gives an induced P4, using an
additional vertex from X5 or X4 creates a vertex of degree 3 in X0 and using an
additional vertex from X0 or X2 creates a vertex of degree 3 in Xi. Because we can
not extend this P4 to an induced P5, we can use at most one additional vertex from
XoUXl
With the above limitations, there are four ways to pick 3 additional vertices to
extend uv to an induced P5 by first picking an additional vertex from X0 or X4 and
then considering whether or not it induces an edge within that Xj. These four induced
P5s are pictured in Figure 3.4. Thus, there are at most tn 4(xmaa;n,)4 Ces in this case.
Figure 3.4: The embeddings of an induced P5 with one funky edge in G[X0,X 1].
Case 2: u G Xi} v G Xj with i j G {2, 4}.
For this case, notice that we cannot pick an additional vertex from Xi without
creating a C3. We examine four subcases based on the number of additional vertices
used in X0. Each of the subsequent possibilities are pictured in Figure 3.5.
If three additional vertices are picked in X0, then we have one way an induced P5
can appear. If two additional vertices are picked in X0, the third additional vertex
must appear in either X2 or X3.
Assume that only 1 additional vertex is picked in Xq. If we pick 1 additional
vertex in X5 (the most we could possibly pick), then the remaining additional vertex
must be picked in either X2 or A3. If we do not pick an additional vertex in X5, then
the two remaining additional vertices must be picked relative to X2, in which 0, 1, or
3
40
2
Figure 3.5: The embeddings of an induced P5 with funky edge in G[X0,Xi\.
2 additional vertices uniquely determine an induced P5.
Finally, assume that no additional vertices are picked in Xq. If X5 has 1 additional
vertex, then picking an additional vertex in X4 forces the remaining additional vertex
to be in X2, otherwise we have a C5. If X4 does not have the second additional vertex,
then we must have at least one additional vertex in X2, otherwise we create a C5.
Thus, there are either one or two additional vertices in X2, both of which uniquely
determine a P5.
The final subcase is if X5 has 0 additional vertices. Again, this uniquely deter
mines a P5 because we must pick 2 additional vertices from the same set to avoid a
C5 and X2 is the only such set where this is possible.
An initial look at counting these Ces gives an upper bound of tn 12(xmaxn)3.
41
However, we can improve this upper bound by accounting for the necessary adjacen
cies when picking at least 2 additional vertices from the same part. Taking this into
account, we have that Figures 3.5a and 3.51 can be in at most tn (Xm*xTlY C3s and
that Figures 3.5b, 3.5c, 3.5f, and 3.5k can be in at most tn (Xmxn^2 xmaxn C3s.
Thus, we have at most
tU (2^7 + 4I + 6) < 7xmaxn4
induced C3s in this case.
Case 3: u Â£ Xi7 v Â£ Xj with i j = 3.
Note that we can pick at most 1 additional vertex in X\ U X2 and X4 U X5,
otherwise we get a C4. Similarly, we can pick at most 1 additional vertex in X\ U X5
and X2UX4, otherwise u or v has at least 3 neighbors. All possible cases are pictured
in Figure 3.6.
Assume we pick the first additional vertex in X3. Then picking the second ad
ditional vertex in X0 forces the third additional vertex to be in either X3 or X4,
both of which give a P5. If we do not pick an additional vertex in X0, then we must
have at least 1 additional vertex in X3, otherwise there is a C4. Having either 1 or 2
additional vertices in X3 both force unique P5s.
Picking the first additional vertex in X5 proceeds as above replacing indices with
their additive inverse in Z,3. Assume that there are no additional vertices picked from
Xi UI5. We proceed by cases of how many additional vertices are picked in Xq. If
3 additional vertices appear in Xq, then the induced P3 is uniquely determined. If
2 additional vertices appear in Xq, then the third additional vertex can be picked in
X2, X3, or X4, each of which determines a unique P5.
If only 1 additional vertex is picked from Xo, then X3 must have at least one
additional vertex, otherwise there is a C3 or C4. Thus, there are three options for the
third additional vertex: X2, X3, and X4, each of which uniquely determines a P5.
42
2
2
2
2
X
X
X
X
(b)
(d)
(i) 0) (k) (1)
(m) (n) (o) (p)
Figure 3.6: The embeddings of an induced P5 with funky edge in G[X0,X3\.
43
If there are no additional vertices picked from Xo, then all 3 additional vertices
must be picked from X3, otherwise there is a C3 or C4. This uniquely determines a
P5
As before in Case 2, we can improve upon the initial count of tn 16(xmaxn)3 by
accounting for when more than one additional vertex is picked from an Xj. Since
Figures 3.6i and 3.6p have 3 additional vertices in the same part and Figures 3.6d,
3.6h, 3.6j, 3.6k, 3.61, and 3.60 have 2 additional vertices in the same part, we have at
most
tU (22^T + 6I + 8) (Xaxn)3 < t 1C)x3maxn4
C6s in this case.
Claim 3.9 There are no funky pairs.
Proof: Suppose for contradiction that there is a funky pair uv with u G X* and
v G Xj with i yt j, Without loss of generality assume u G X0. Let G' be the graph
obtained from G by fixing the funky pair uv; that is, making uv an edge if j G {1,5}
or making uv a nonedge if j G {2,3,4}. We compare the number of induced Cqs
containing {u,v} in G and in G' to obtain a contradiction.
In G', picking one vertex from each X& with k G {1,..., 5} \ {j } forms an induced
Ce as long as none of the resulting 14 pairs is funky. If one of the resulting pairs is
funky and incident to uv, say uv', then we have at most
max{xiX2X3, X1X2X4, X1X2X5, X1X3X4, X1X3X5, X1X4X5, X2X3X4, X2X3X5, X3X4X5}n3
induced Cqs by picking the three remaining vertices from distinct X^s among the three
XjS not containing u,v,v'. If one of the resulting pairs is funky and not incident to
uv, say u'v', then we have at most
fn raax{x\X2, XiX3,X\X^,XiX3,X2X3, X2X4, X2X5, X3X4, X3X5, x^x^n2
44
induced C^s by picking the two remaining vertices from distinct Xs among the two
XiS not containing u,v,u',v'. Thus G' has at least
p^min ^ fiTiax%max I'1'mn.r\ ^ (3'7)
induced Ces containing uv.
For comparison, we count the number of induced Ces in G containing uv. Claim
3.7 guarantees any such Ce includes a second funky pair. The number of Ces contain
ing another funky pair u'v' with {u, v}r\{u', v'} = 0 can be generously upper bounded
by picking any funky pair and two additional vertices, that is (fn2)n2. Thus, in the
subsequent bounds we can assume that all funky pairs are incident to uv.
If there are at least two such funky pairs in an induced Ce, then we can generously
upper bound the number of Ces by assuming every such pair of incident funky edges
creates a Ce. That is, there are at most 2 (jes with at least two funky
pairs incident to uv.
If there is only one funky pair incident to uv in an induced Ce, we claim that
either the Ce includes a vertex from T or the choice of the remaining 3 vertices in the
Cq are forced. In the first case, we can generously upper bound the number of such
Ces by picking an incident funky edge, a vertex from T, and then any two additional
vertices. This gives an upper bound of (df(v) + df(u))n tn n2.
Let C be an induced Ce containing uv that does not include a vertex from T and
has only one funky pair incident to uv. Since we previously counted all of the Ces
with a funky edge not incident to uv, we can assume that all (both) funky pairs in
C are incident to u, say uv and uv1 are the two funky pairs of C. Without loss of
generality, let u G Xj and v G X, with i yt j. Note that C u is an induced P5.
Thus, if v' G Xj, then V(C u) C Vj. Consequently, the number of choices for C
is upper bounded by df(u)n (Xjn)3, which we will later double to account for the
choice between u and v.
On the other hand, assume that v' G Xk with 0 ^ k ^ j. We will show that the
45
choices for the remaining three vertices of C are forced into distinct XjS, which gives
an upper bound of df{u)n (xmaxn)3. The argument for this upper bound is presented
in the following three paragraphs should the trusting reader want to reference Figure
3.7 and skip the case analysis.
Figure 3.7: Induced C^s with all funky edges incident to a single vertex.
First, assume u,v E Xg. Since C u is an induced P5 with u incident to its
endpoints and v' not in the same Xj as v, then v must be an internal vertex of this
path. Because two consecutive vertices of an induced P5 in the same Xj would create
a C3, each vertex of this P5 must be in distinct XjS. Considering v is an internal
vertex, there are three potential options for the choice of P5: V0VV2V3V4, V5V0W2V3,
and V4V5V0W2, where vg E Xg. Notice that the third possibility would require an
additional funky pair for an induced P5. Hence, we do not include it in this count.
The choice of the funky pair incident to u giving rise to C gives v' in either X3 or
X4, which determines the choices of the remaining three vertices into X5,Xo,X2 or
X0,X2,X3, respectively. A similar argument for v E X5 holds by replacing indices
with their additive inverse in Z6.
46
Next, assume u E X0 and v E X2. Since uv is an edge, u must be an endpoint
of the induced P5 C u. As before in regard to consecutive vertices, C u spans
either X2X3X4X5X0 with v' E X5, or X4X5X0X1X2. Note that the second option is
not possible as it requires three funky pairs be incident to u. Thus, the choice of a
funky pair incident to u giving rise to C gives v' E X5, which determines the choices
of the remaining three vertices. As before, a similar argument for v E X4 holds by
replacing indices with their additive inverse in Z6.
Finally, assume u E X0 and v E X3. As uv is an edge, v must be an endpoint of
the induced P5 C u. As in the two previous cases, C u spans either X3X4X5X0Xi
with v' E X5 or X3X0XiX2X3 with v' E X3. Thus, the choice of a funky pair incident
to u giving rise to C gives v' E X1UX5 and the choices for the remaining three vertices
are forced into the appropriate X^s.
We now consider the remaining two cases when there are no funky pairs incident
to uv, both of which use vertices from T. If there are at least two vertices from T,
then we can generously upper bound the number of induced C3s containing uv by
picking two vertices from T and any two additional vertices. This gives at most
such CqS. If there is precisely one vertex from T in an induced C3 containing uv, then
Claim 3.8 gives tn 10(xmaxn)3 as an upper bound.
Combining and comparing all of the counting bounds within this claim, the num
ber of induced Ces containing uv divided by n4 is
inG: <
2 df
2
+ / + 2 df(xliax
+ t) + lOtx
3
max
t
< 0.000505
in G xmin 2dfXmax
dfX
2
max
> 0.000630.
This contradicts the extremality of G.
We have just shown that there are no funky pairs among (J Xj. In further cleaning
the top level structure of G, we want to show that T = 0. Once completed this will
show that the vertices of G can be partitioned into 6 sets that resemble the blowup
47
of a Ce
Suppose there exists x E T. We will move x to one of the X* such that df(x)
is minimal. By symmetry we may assume that x is added to Xq. Note that adding
a single vertex to X0 does not change any of the density bounds we used above by
more than o(l).
Claim 3.10 For every x eT if x is added to X0, then df(x) > 0.0481706.
Proof: Let xw be a funky pair. Let G' be obtained from G by fixing xw through
adding or deleting the edge xw. Since G is extremal, we have C(G') < C(G). The
analysis for this proof is similar to that in Claim 3.9. However, we can say a bit more
since every funky pair contains x.
First, we count the induced Cqs containing xw in G. We generously give an
upper bound for the number of CqS containing xw and another vertex from T by
selecting a vertex from T and any three remaining vertices, that is (tn) n3. As in
Claim 3.9, any induced 6cycle C containing x has C x as an induced P5. Citing
the previous analysis and Figure 3.7, the number of induced C5S containing xw is at
most (df{x)n) (xmaxnf.
In G' there are at least (xminn)4 (df{x)n) (xminn)3 induced C6s containing xw.
Thus, we have
C(G)
n4
C(G')
n4
Since C(G') < C(G), we have
< df(x)x3max + t, and
A 'X'min df(x')Xmin
min df (x)Xmin A df{x)Xmax T t,
which together with (3.3) and (3.5) gives df(x) > 0.0481706.
For vertices u,v E V(G), let Cg denote the number of Cbs in G containing u and
Cqv denote the number of Cqs containing both u and v.
48
Claim 3.11 Every vertex of the extremal graph G is in at least (y^ + o(l)) ()
0.000128617n5 induced Cqs.
Proof: Note that a trivial bound is Cfv < (42). Consider G' obtained from G by
deleting v and adding a copy of u. Since G is extremal, we have
Thus, Q Cl< n4.
Suppose for contradiction that a vertex of G is in less than o(l)) ()
induced Cgs. Since Cf Cq < n4 for all u, v G V(G), every vertex of G is in less than
Having established the number of induced C6s every vertex of G is in, we aim to
show that any vertex of T is in too few. Once this is shown, G will have no funky
Claim 3.12 T is empty.
Proof: Assume x E T. We count the number of induced C6s containing x. Let a^n
be the number of neighbors of x in Xi and hpn be the number of nonneighbors of x
in
The number of C6s containing x and five vertices from (J Xi is upper bounded by
where we count induced P5s when vertices are in distinct parts and when vertices are
all in the same part. The variables a*, 6* satisfy the upper bound in (3.3) and Claim
(lifeod)) + n4 induced C6s. Summing over all vertices of G and dividing by 6
to adjust for overcount, we have that G contains less than
induced C6s. Since CgX has + o(l)) (g) induced Cgs, G is not extremal, a
contradiction.
pairs and T = 0. Note that this will roughly determine the top level structure of G.
3.9.
49
Moreover, we also need to include the cases that the C^s can contain vertices from
T. To this end, let an be the number of neighbors of x in T and bn be the number of
nonneighbors of x in T. We generously upper bound the number of Cqs containing
vertices of T by increasing each of the cps and biS by a and 6, respectively. Thus, we
want to solve the program
maximize Ei(a* + a)(&i+i + b)(bi+2 + b)(bi+3 + b)(ai+A + a) + ^ a\b\,
subject to a + b + + &*) = !,
(P)
0.16583
a + b < 0.00031,
bi + ai+i + a%12 + Hi13 + bl+4 > 0.048,
ai,bi,a,b > 0,
in which the penultimate constraint uses Claim 3.10 in lower bounding the funky
degree of x. Instead of solving (P), we solve a slight relaxation (P') with increased
upper bounds on ai + bi, which allows us to drop a and b. Since the objective function
is maximizing, we can claim that + bi is always as large as possible.
maximize
h = Y, aibi+ibi+2bi+3ai+4: + ^ E a?6f,
(P')
subject to ai + bi = 0.16751 + 0.00031,
<
bi + ai+i + Oi12 + cp+3 + bi\4 > 0.048,
di,bi > 0.
Note that the resulting program (P7) has only 6 degrees of freedom.
We find an upper bound for the solution of (P7) through a naive optimization.
We first discretize the set of feasible solutions with a uniform mesh and then bound
the gradient of the objective function h to account for the behavior between mesh
points. For this, we fix a constant s that will correspond to the number of steps in
each variable from 0 to 0.1679 when generating our uniform mesh, which include the
50
endpoints. Through its construction, this mesh includes all feasible solutions of (P'),
and hence of (P), in one of the 6dimensional mesh boxes.
After generating our mesh, we must find the partial derivatives of h. Since h is
symmetric, we only check the partial derivative with respect to ao
= bib2b3a4 + a2hb4h + (Uo&o
oao 6
We want to find an upper bound on Jy. Picking 0.1679 as an upper bound for a* + 6*
allows us to assume
&o T bo a2 T b2 <^3 + 63 CI4 + 64 b\ 65 0.1679,
while we maximize
bib2b3a4 + (Z2&3&4&5 = 0.1679(0.1679 a3) [(0.1679 a2)a4 + a2(0.1679 a4)]
= 0.1679(0.1679 a3)(0.1679 a2 2a2a4 + 0.1679 a4).
This is maximized if a2 = 0,a4 = 0.1679 or a2 = 0.1679, a4 = 0 and gives the value
0.16794. Hence,
a0&o = 27a0 ( ^ ] <
b0 \ 3 ^ 27(a0 + b0)4 27 0.16794
3"
44
256
The resulting upper bound is
< 0.1679* + < 0.0008283.
oa0 5 256
Hence, in a 6dimensional mesh box with side length d, the value of h cannot be bigger
than the value at a corner plus y 0.0008283, where the factor y comes from the fact
that the closest mesh point is a distance at most  in each of the 6 variables.
When s = 80, we compute the maximum over all mesh points to be at most
0.00009983. Along with the previous arithmetic computations for bounds, this can
be checked using the C++ code provided in a supplemental hie named meshopt. cpp.
With t = '4q79 we have
6d
0.0008283 < 0.0000052152.
51
Therefore, we conclude that x is in at most
(0.00009983 + 0.0000052152)n5 < O.OOOlln5
induced C6s, a contradiction with Claim 3.11.
As a brief recap, we have shown that the vertex set of of G can be partitioned
into 6 sets that resemble the blowup of a Ce. With no funky pairs between these
sets, we have nearly determined that the top level structure of the extremal graph G.
All that remains to show to prove Theorem 3.4 is to show that the sets are balanced.
To this end, notice that induced Cqs can only appear within this structure either
by intersecting each Xi exactly once or by being entirely contained within a single
Xi. This implies that
C(n) = n6 \\xi + ^2 c(xin),
i i
where C(n) denotes the maximum number of induced Cqs in a graph of order n. A
direct translation of Proposition 3.1 gives
C(n + 1)
ra+ly
6
for all n. Since C(n) > 0,
exists. Thus,
t :
lim
nCO
t + o(l) = 6! Xi +1 ^2 x\i
i i
which, with the constraints on ay, gives ay =  + o(l) and t
show that \Xi\ \Xj\ < 1.
(3.8)
24
1555'
It remains to
Claim 3.13 For large n, we have Xj \Xj\ < 1 for all i, j E Z6.
Proof: Suppose for contradiction that IX^ \X2\ > 2. Pick u E Xi so that Cf is
minimized over vertices in Xi and pick v E X2 so that Cg is maximized over vertices
52
in X2. Since G is extremal, C% + C%v Cl > 0 otherwise the number of C^s can be
increased by replacing u by a copy of v.
Dehne y^ := \Xi\ = Xirn. By (3.8), we have
J! ,,m > Â£M > Â£M > Ji_
1555 1 () () 1555'
Using that y\ y2 > 2 and that removing a vertex from any induced Cq gives an
induced P5 either intersecting exactly 1 part 5 times or intersecting 5 parts exactly
once,
Ce + CT C% <
<
C{yi) C(y2)
1 y2ysy4y5ye + ymy^y&ymymye
y 1
V2C(yi) yiC(y2)
ym
<
<
24
1555
24
1555 6!
24
1555 6!
= ijji ~ Z/2)
, 224
<
o(l)
y 2
i]j2 ~yi + 1)2/32/42/52/6
y2
yi
6
{y2 yi + ^)ymy^y&
+ 0(1)J (yi ya) + (y2 yi + ^yaymye
+ 0(1)) (yi y2)(y^ + y\y2 + ywl + y2)
(
(y2 yi + ^)yzymy&
24
V1555 6!
o(l)
4n4 nA \ (1 + o(l))n4
1296 1296 ) + 1296
1555 6!
0(1) 1 XL liihhhl < 0,
1296
1296
a contradiction with Cq + Cqv C% > 0.
This claim forces the six X^s, which resemble the blowup of a Ce, to be balanced.
Hence, the proof of Theorem 3.4 is complete.
3.2.2 Iterative Structure of the Extremal Graph
Having completed the stability portion of the iterated approach that determines
the top level structure of the extremal graph, we turn our attention to inductively
proving that the extremal graph has the desired iterated structure. As such, Theorem
3.3, which states that CqX is the unique extremal construction for a graph on 6fc
53
vertices, is a consequence of Theorem 3.4. For the proof of Theorem 3.3 we will
take a minimal counterexample and show that a blowup of this graph contradicts
Theorem 3.4.
As previously mentioned, C\x is simply Co Thus, Theorem 3.3 is clearly true for
k= 1.
Suppose for contradiction that there is a graph G on n = 6k vertices with C(G) >
C(Ckx ) that is not isomorphic to Ck x. Fix k > 2 to be minimal in that every extremal
graph on 6fc_1 vertices is isomorphic to C'gfc1'>x.
If G conforms to the structure described in Theorem 3.4, then each extremal
part on 6fc_1 vertices is isomorphic to C'gfc1'>x by the minimality of k. This makes G
isomorphic to Ckx. Therefore, V(G) can not be partitioned into sets X* for i e Z6
with \Xi\ = 6fc_1 and the cyclic adjacency structure described in Theorem 3.4.
In relation to Theorem 3.4, take no to be sufficiently large and pick l so that
6e > no Let H be an extremal graph on 6e vertices. We construct two graphs on
6k+i vertices using H and will rely on the structure of H given by Theorem 3.4 to
compare them. Construct G\ from Ckx by blowing up every vertex into a set of 6e
vertices and insert a copy of H into each. Through Â£ applications of Theorem 3.4, G\
is an extremal graph. For comparison, construct G2 from G in the same way. Since
C(G) > C(Ckx),
C(G2) C{Gi) = (6fc C{H) + C{G) (6^)6) (6fc C{H) + C(Gkx) (6^)6)
= (6e)6 (G(G) G(Gkx))
> 0.
Thus, C(G2) > C(Gi). Hence, G2 must also be extremal. Since G2 has 6k+e > 6k > no
vertices, Theorem 3.4 guarantees that V(G2) can be partitioned into sets X0,... ,X5
with \Xi\ = Qk+l~l and the described adjacency structure.
We take a moment to consider such a partition. If u,v E Xi} then N(u) and
54
N(v) both contain Xi+i U Xj+5, and neither intersect Xi+2 U Xi+3 U Xi+4. Thus the
adjacency pattern, that is the neighbors and nonneighbors, of u and v agrees on more
than  of V(G2) \ {u, v}, and has the potential to disagree only in Xj. There are three
cases to consider if u Â£ Xi and v G Xj with i yt j. First, if u G X* and v G Xi+4, then
the adjacency pattern of u and v agrees on Xi+3 U Xi+4 and disagrees on Xi+2 U Xi+5.
Next, if u G Xi and v G Xi+2, then the adjacency pattern of u and v agrees on
Xi+4 U Xi+4 and disagrees on Xi+3 U Xi+5. Lastly, if u G Xj and v G Xi+3, then
the adjacency pattern of u and v disagrees on Xi+4 U Xi+2 U Xi+4 U Xi+5. Therefore,
if u G Xi and v G Xj with i yt j, then the adjacency pattern of u and v agree on
less than  of V(G2) \ {u,v}. Therefore, a pair of vertices in G2 are in the same Xj
precisely when their adjacency pattern agrees on more than  of V(G2) \ {u,v}.
However, consider the construction of G2. The adjacency pattern for any pair of
vertices in a copy of H agrees on G2 H, which is
6k+e 6e ^ gfc x ^ 35 ^ 3
6k+e ~ 6k ~ 36 > 4
since k > 2. This implies that the vertices for each copy of H in the blowup of G are
in the same Xi} that is V(H) C Xj for a fixed copy of H and some Xj. Recalling that
each vertex of G corresponds to a copy of H, the partition on V(G2) in which each
copy of H is in the same Xj induces a partition of V(G). Because this partition on
V(G2) has the structure described in Theorem 3.4, we have a partition of V(G) with
the structure described in Theorem 3.4. The existence of this partition contradicts
our initial choice of G and completes the proof of Theorem 3.3.
55
COMPUTATIONS IN GRAPH COLORING
Graph coloring is an immensely popular area of research in graph theory. This area, at
its core, partitions a graph so that the representative parts satisfy a given property.
The classic graph coloring problem, vertex coloring, partitions the vertex set into
independent sets. One of the most common problems in graph colorings, the Four
Color Theorem, has significant impact on the work appearing in this part, which
is joint with Jennifer Diemunsch and Sogol Jahanbekam. The author contributed a
key component of the main results by proving Lemma 5.11, computationally verified
coefficients of polynomials in Lemmmas 5.12, 5.13, and 5.14 to allow application of the
Combinatorial Nullstellensatz, and actively contributed in discovering the discharging
rules integral to the proofs in Section 5.3.
56
CHAPTER 4
AN INTRODUCTION TO GRAPH COLORING
We begin with some basic definitions. A vertex coloring of a graph G is a function
4>: V(G) > C, where C is a set of colors. The vertex coloring is proper if every
pair of adjacent vertices are assigned distinct colors. A kcoloring is a vertex coloring
that uses at most k colors. A graph is kcolorable if there exists a proper fccoloring
on its vertex set. The chromatic number of a graph G, denoted x(G), is the smallest
nonnegative integer k such that G is fccolorable. In the broader context of graph
coloring viewed as partitioning the underlying graph structure, vertex coloring parti
tions the vertices of the graph into independent sets, each of which is referred to as
a color class.
Karp [65] proved that determining the chromatic number of a graph is NP
complete, which means it is possible to quickly (in polynomial time) check a given
solution but it is not currently known how to quickly find the solution. Moreover, if
one could find a quick solution, then a large number of other problems believed to be
very difficult could be solved quickly as well. Although computational approaches to
determining the chromatic number of a graph may not be efficient, there are many
theoretical bounds.
The independence number of a graph G, denoted a(G), is the size of a maximum
independent set in G. This definition gives the following immediate bound:
X(G)>
\V(G)\
a(G) '
While this bound is rather naive, Bollobas [15] showed it to be very close for almost
all graphs.
The presence of edges both globally and locally have been used to bound the
chromatic number of a graph. From a global perspective, Nordhaus and Gaddum [71]
showed for any graph G and its edgecomplement G, x(G) + x(G) A U(G) + 1. In
57
a local perspective, Brooks [23] showed that if G is a connected graph then x(G) <
A(G) unless G is complete or an odd cycle.
Characteristics of vertex colorings have also been studied. For example, Hajnal
and Szemeredi [51] showed that every graph G admits a (A(G) + l)coloring such
that each color class has
A(G)I
A(G) + 1
or
A(G)I
A(G)+1
vertices. More generally, Kierstead
and Kostochka [67] showed that if d(u) + d(y) < 2D + 1 for every xy E E(G), then G
admits a (D + l)coloring such that each color class has
A(G)I
D+1
or
A(G)I
D+1
vertices.
4.1 The Four Color Theorem and the Discharging Method
The chromatic number of many families of graphs are known. For example, a
graph G has chromatic number x(G) = 1 if and only if G has no edges. A graph
G has chromatic number x(G) < 2 if and only if G is bipartite. In particular, trees
are connected graphs with no cycles and forests are graphs in which every connected
component is a tree. The nice structure of trees makes it easy to show that trees, and
subsequently forests, are bipartite.
Additionally, for any k E N, x(^2k) = 2 and x(C2fc+i) = 3. The girth of a graph
is the length of its shortest cycle. It is interesting to note that although a graph G
with large girth may look very much like a tree in smaller areas, it is possible for G
to have an arbitrarily large chromatic number [32],
Another family of graphs with nice structure is planar graphs, which can be drawn
in the plane with no crossing edges. When discussing planar graphs, we will often
refer to the faces of the graph, which are regions in the plane bounded by a closed
walk. For a plane graph G, we will let F(G) denote the set of faces of G. A plane
graph in which all faces have length 3 is called a triangulation. The nice structure of
planar graphs gives rise to the following equation:
Proposition 4.1 (Eulers Formula [39]) For a connected, plane graph G,
\V(G)\\E(G)\ + \F(G)\ = 2.
58
The following is a straightforward application of Eulers Formula:
Proposition 4.2 A triangulation G satisfies if(G) = 3E(G) 6.
Proposition 4.3 For a plane graph G,
Y Kf) = mm
feF(G)
where 1(f) represents the length of face f in G.
While true for all graphs, not just planar graphs, we include the following here
for ease of reference:
Proposition 4.4 (Degree Sum Formula) For any graph G,
Y d(v)=2\E(G)\,
vev(G)
where d(v) represents the degree of vertex v in G.
With the arbitrarily large gap possible between girth and chromatic number in
mind, the following is perhaps somewhat surprising.
Theorem 4.5 (The Four Color Theorem [5, 6]) If G is a planar graph, then
X(G) < 4.
Conjectured by Guthrie in 1852, the Four Color Theorem took over 100 years to
prove. Along the way many false proofs were announced, most notably of Kempe
[66] in 1879 and Tait [87] in 1880. Fleawood [56] showed that every planar graph is
5colorable, and Grotzsch [48] showed that every planar graph with girth at least 4
is 3colorable. For a survey of results on coloring of plane graphs, see [21]. Before
proceeding, it is worth mentioning both that Fladwigers Conjecture [50] is true for
4chromatic graphs [31], 5chromatic graphs [94], and 6chromatic graphs [80], and
also that each proof uses the Four Color Theorem.
59
One of the most significant outcomes from efforts to prove the Four Color Theorem
is the development of the Discharging Method. The general approach to this method
can broadly be described in two steps. First, global properties are used to guarantee
the existence of local configurations, which are often referred to as unavoidable
configurations. Then, these local configurations are shown not to exist in a minimal
counterexample to the desired conclusion, that is the configurations are reducible.
Thus, the goal of many proofs is to find a list of unavoidable configurations and show
that all of those configurations are reducible.
Proving that unavoidable configurations are reducible often involves detailed case
analysis. In some instances, the case analysis involved is wellsuited for computer as
sistance. For example, Appel and Hakens proof of the Four Color Theorem [5, 6]
used computers to help confirm the reducibility of 1,936 configurations under 487 dis
charging rules. Widely touted as the first computerassisted mathematical proof, the
validity of the result has withstood much skepticism. In 1997, Robertson, Sanders,
Seymour, and Thomas [79] presented a proof using only 633 reducible configurations
and 32 discharging rules. Although needing significantly fewer reducible configura
tions, the proof still utilized computer assistance. Further exposition about the Four
Color Theorem and its proof can be found in an article by Woodall [96] and a book
by Wilson [95].
To discover unavoidable configurations, charge is assigned to elements of a graph
and then reallocated around the graph under a set of rules, that is discharging rules.
To discover unavoidable configurations in the following example that appears in [19],
we need the following definition: a fcedgeconnected graph is essentially {k + 1)edge
connected if all of its fcedge cuts are associated with a single vertex. As an immediate
consequence of this definition, the minimum degree of an essentially fcedge connected
graph is k 1.
Proposition 4.6 Every essentially 6connected triangulation G must contain a ver
60
tex of degree 5 adjacent to a vertex of degree 5 or 6.
Proof: Assign each vertex v a charge of 6 d(v), where d(v) represents the degree
of v. From Propositions 4.4 and 4.2, the total charge of the graph is
Â£ (6)=6r(G)2Â£(G)=12.
vev{G) i>ev(G)
Our aim in discharging is to reallocate charge so that every vertex has nonpositive
charge, which is impossible as the total charge is 12 and no charge is destroyed. Notice
that vertices of degree at least 6 are given a nonpositive initial charge. Since G is an
essentially 6connected triangulation, it has minimum degree 5. Thus, we need only
distribute charge away from vertices of degree 5 without giving away so much charge
that other vertices end up with positive charge.
Consider the following reallocation of charge: for every vertex of degree 5, dis
tribute its 1 charge equally among its 5 neighbors. Every vertex v can receive at most
\d{v). Thus, a vertex of degree at least 8 has a final charge of
1 4 32
6 d(v) + d(v) = 6d(v) <6<0.
5 5 5
Similarly, each vertex of degree 7 with at most 5 neighbors of degree 5 has nonpositive
final charge, as do vertices of degree 5 or 6 with no neighbor of degree 5. Thus, G
must have either a vertex of degree 5 adjacent to a vertex of degree 5 or 6, or a vertex
of degree 7 with 6 or 7 neighbors of degree 5.
However, since G is a triangulation, consecutive neighbors of a vertex are adjacent.
Thus, the neighborhood of a vertex of degree 7 with at least 4 neighbors of degree 5
has a pair of adjacent vertices of degree 5. Hence, it is enough to say that G must
have either a vertex of degree 5 adjacent to a vertex of degree 5 or 6.
Discharging arguments appear in results related to a 1975 conjecture of Steinberg,
which states that planar graphs without 4 or 5cycles are 3colorable. Note that this
conjecture has very recently been disproved by CohenAdda, Hebdige, Krai, Li, and
61
Salgado [27]. A particularly nice discharging argument appears in Abbott and Zhou
[1] regarding the 3colorability of planar graphs with no fccycles for 4 < k < 11, which
was improved by Borodin, Glebov, Raspaud, and Salavatipour [22] to 4 < k < 7.
Cranston and West [28] provide a friendly, comprehensive guide to the discharging
method that includes a variety of examples and existing results. Salavatipour [81]
provides a nice survey of discharging results in his dissertation.
Our results center around a specific coloring of planar graphs and use computer
assistance to prove that certain configurations are reducible. Rather than checking
long lists of detailed cases to do so, we embed information relevant to our coloring into
polynomials and then combine computing power with the theoretical result presented
in the following section.
4.2 List Coloring and the Combinatorial Nullstellensatz
A natural generalization of vertex coloring is list coloring. While a vertex coloring
colors each vertex from a single pallet of colors, every vertex is colored from its own
pallet of colors in a list coloring. More formally, a list assignment on a graph G
is a function L: V(G) > 2R. A graph G is Lcolorable (or list colorable when the
context is clear) if G admits a proper vertex coloring (f) such that 4>{v) G L(v) for all
v G V(G). A graph is kchoosable if it is Lcolorable for every list assignment L with
\L(v)\ > k for all v G V(G). The list chrom,atic number of G, denoted xr(G), is the
smallest k G N such that G is fcchoosable.
There is a vast collection of work rooted in comparing the list chromatic number
of a graph to its chromatic number. To mention a few, Erdos, Rubin, and Taylor
[36] showed that these two graph parameters can be arbitrarily far apart. For any
graph G, they also showed that xr(G) + Xr(G) R(G) + 1 and Xr(G) A(G), the
upper bounds of which match upper bounds for x(G). In relation to the Four Color
Theorem, Thomassen [88] showed that every planar graph is 5choosable and Voigt
[92] showed that there exist non4choosable planar graphs. In relation to Grotzschs
62
Theorem [48], Kratochvll and Tuza [68] showed that every planar graph of girth at
least four is 4choosable, Thomassen [89] showed that every planar graph of girth
at least five is 3choosable, and Voigt [93] showed that there exist non3choosable
planar graphs with girth at least four.
A useful tool for list coloring and list versions of other graph colorings is Alons
Combinatorial Nullstellensatz. Application of this useful theorem has occurred in
mathematical fields other than graph theory and combinatorics such as additive
number theory and discrete geometry. Formally stated below, the Combinatorial
Nullstellensatz provides criteria to guarantee the existence of a nonzero solution to a
multivariate polynomial from a collection of subsets of a held.
Theorem 4.7 (Combinatorial Nullstellensatz [3]) if nil ;1 xf is a monomial
with nonzero coefficient in a polynomial f having degree Ym=\ h over a fie^ and
Si,... ,Sn are sets with Si > U for i E [n], then f(x) f 0 for some x E n!=i Si
lo. applying the Combinatorial Nullstellensatz to reducibility arguments within
discharging proofs, we construct coloring polynomials whose zeros correspond to
breaking the rules of the desired coloring. After computationally verifying the exis
tence of a monomial having maximum degree in these coloring polynomials, we can
apply the Combinatorial Nullstellensatz to obtain a nonzero solution. A nonzero
solution to a coloring polynomial corresponds to a proper coloring of the graph con
figuration used to construct the coloring polynomial. These graph configurations are
in fact the reducible configurations for our results.
Working within a minimal counterexample supposedly containing such a con
figuration, we delete portions of the configuration, inductively color the remaining
graph, and then construct a coloring polynomial that reflects the restrictions of ex
tending the inductive coloring to the entire minimal counterexample. Application
of the Combinatorial Nullstellensatz to the constructed coloring polynomial guaran
63
tees the existence of a proper coloring on the minimal counterexample containing the
configuration, which makes the configuration reducible since the minimal counterex
ample cannot be properly colored. This application is precisely where our arguments
avoid much of the detailed case analysis often present within reducibility arguments
for discharging.
64
CHAPTER 5
LUCKY LABELING
There are a multitude of variations on vertex coloring, see Gallians dynamic
survey [46]. We consider a derived vertex coloring in which each vertex receives a
color based on assigned labels of its neighbors. Let t: V(G) > E be a labeling of the
vertices of a graph G. For each v G V(G), let Sg(v) = Â£(u), where NG(v) is
ueNa(v)
the open neighborhood of v in G. When the context is clear, we use S(v) in place of
SG(v). We say the labeling t is lucky if for every pair of adjacent vertices u and v, we
have S(u) yt S(v), that is S induces a proper vertex coloring of G. The least integer
k for which a graph G has a lucky labeling using labels from {1,,k} is called the
lucky number of G, denoted rj(G).
Determining the lucky number of a graph is a natural variation of a wellstudied
problem posed by Karonski, Luczak, and Thomason [64], in which edge labels from
{1,,k} are summed at incident vertices to induce a vertex coloring. Karonski et
al. conjecture that edge labels from {1,2,3} are enough to yield a proper vertex
coloring of graphs with no component isomorphic to JW This conjecture is known
as the 1,2,3Conjecture and is still open. In 2010, Kalkowski, Karonski, and Pfender
[63] showed that labels from {1,2, 3, 4, 5} suffice.
Similar to the lucky number of a graph, Chartrand, Okamoto, and Zhang [25]
defined cr(G) to be the smallest integer k such that G has a lucky labeling using k
distinct labels. They showed that cr(G) < y(G). Note that cr(G) < rj(G), since with
rj(G) we seek the smallest k such that labels are from {1,..., k} while allowing some
integers in {1 to not be used as labels, whereas cr(G) considers the fewest
distinct labels without concern for the value of the largest label.
In 2009, Czerwinski, Grytczuk, and Zelazny proposed the following conjecture for
the lucky number of G:
65
Conjecture 5.1 (Czerwinski, Grytczuk, Zelazny [29]) For every graph G, r/(G) <
X(G).
This conjecture remains open even for bipartite graphs, for which no constant
bound is currently known. Czerwinski et al. [29] showed that rj(G) < k + 1 for every
bipartite graph G having an orientation in which each vertex has outdegree at most
k. They also showed that rj(G) < 2 when G is a tree, rj(G) < 3 when G is bipartite
and planar, and r/(G) < 100,280,245,065 for every planar graph G. Note that if the
conjecture is true, then r/(G) < 4 for any planar graph G. The bound for planar
graphs was later improved to rj(G) < 468 by Bartnicki et al. [13], who also show the
following.
Theorem 5.2 (Bartnicki et al. [13]) If G is a 3colorable planar graph, then
rj(G) < 36.
The girth of a graph is especially useful in giving a measure of sparseness. Know
ing the girth of a planar graph gives a bound on the w,axiw,uw, average degree of a
graph G, denoted mad(G), which is the maximum average degree over all subgraphs
of G. The following proposition is a simple application of Eulers formula (Proposition
4.1 and gives a relationship between these two parameters:
Proposition 5.3 If G is a planar graph with girth g, then mad(G) < jfk,.
Proof: Let H C G. Note that the girth of H is at least g. Beginning with the
average degree of H and applying Eulers formula (Proposition 4.1), we have
E6V(H) d(v) 2\E(H)\ _ 2\E(H)\
\V(H) 2 + Â£(J/)F(ff) < Â£(ff) F(J/)r
Note that each face has at least g edges. Because each edge is counted exactly twice
in summing the lengths of faces in a planar graph, g\F(H)\ < 2\E(H)\. Thus,
E V(H) g 2\E(H)\ 2g\E(H)\ 2g
\V(H)\ g \E(H)\ \F(H)\ ~ g\E(H)\ 2\E(H)\ g2'
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By the arbitrary choice of H, every subgraph of G satisfies the above inequality, which
gives the desired result.
Bartnicki et al. [13] proved the following:
Theorem 5.4 (Bartnicki et al. [13]) If G is a planar graph of girth at least 13,
then rj(G) < 4.
In 2013, Akbari, Ghanbari, Manaviyat, and Zare [2] proposed the list version of
lucky labeling. A graph is lucky kchoosable if whenever each vertex is given a list of
at least k available integers, a lucky labeling can be chosen from the lists. The lucky
choice number of a graph G is the minimum positive integer k such that G is lucky k
choosable, and is denoted by qi(G). Akbari et al. [2] showed that qi(G) < A2 A + 1
for every graph G with A > 2. They also proved the following:
Theorem 5.5 (Akbari et al. [2]) If G is a forest, then qe(G) < 3.
In this paper, we improve these results for planar graphs of particular girths.
Specifically, we use the Combinatorial Nullstellensatz within reducibility arguments
of the discharging method to prove our results. The combination of these two popular
techniques is a novel approach that can eliminate a considerable amount of case
analysis. Moreover, using the Combinatorial Nullstellensatz in reducibility arguments
of coloring problems enables proving choosability results, rather than just colorability.
We show the following improvements on the lucky choice number for planar graphs
of given girths:
Theorem 5.6 Let G be a planar graph with girth g.
1 If 9 A 5, then qi{G) < 19.
2. If g > 6, then qi{G) < 9.
3. If g > 7, then qe(G) < 8.
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4 If 9 ^ 26, then rje(G) < 3.
Various 3colorings of planar graphs have been obtained under certain girth as
sumptions. Combined with Grotzschs result [48], our result answers Conjecture 5.1
for nonbipartite planar graphs with girth at least 26.
In Section 5.1, we introduce the notation and tools that are used throughout the
remainder of the chapter. We also give an overview of how we use the discharging
method and the Combinatorial Nullstellensatz. Section 5.2 describes certain reducible
configurations. Finally, in Section 5.3 we prove Theorem 5.6.
5.1 Notation and Tools
Let Ng(v) be the open neighborhood of a vertex v in a graph G. For convenience,
a jvertex, j~vertex, or j+vertex is a vertex with degree j, at most j, or at least
j, respectively. Similarly, a jneighbor (respectively j~ neighbor or j+neighbor) of
v is a jvertex (respectively j_vertex or j+vertex) adjacent to v.
For sets A and B of real numbers, A B is defined to be the set {a + b: a E
A,b E B}. Likewise, A 0 B is defined to be the set {a b: a E A,b E B}. When
5 = 0, we define AB = AqB = A. We use the following known result from
additive combinatorics:
Proposition 5.7 Let A\,... ,Ar be finite sets of real numbers. We have
r
\A\ Ar > 1 + 'y ^ (\Ai\ 1).
i= 1
r r
Proof: We apply induction on Yf, IAi\ When Yf A = 1, all but one Ai are empty,
i= 1 i= 1
so we have \A\ Ar\ = 1, as desired.
r
Now, suppose that Yf, I^M = n We may suppose that all Ai are nonempty.
i= 1
Let ^ be the minimum element of Ai for i E {1,... ,n}. Let A[ = Ai {ai}. By
r
the induction hypothesis, we have \A\ A2 Ar\ > 1 + Yf, (1^1 1) 1.
i= 1
However, \Ai Ar\ > {ai + + ar} + \A\ A2 Ar\. Therefore,
A\ Ar > 1 + Yf, (^4j 1) b
i= 1
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Note that A(Â£?) is the same as AQB, where B = {b : b G B}. This yields
the following known corollary:
Corollary 5.8 Let A and B be nonempty sets of positive real numbers. We have
\AQB\ > \A\ + \B\ 1.
Throughout, we consider when endpoints of edges need different sums to yield a
lucky labeling. For this reason, if we know S(u) S(v) for an edge uv of G, then we
say that uv is satisfied; uv is unsatisfied otherwise.
Our proofs rely on applying the discharging method. This proof technique assigns
an initial charge to vertices and possibly faces of a graph and then distributes charge
according to a list of discharging rules. A configuration is kreducible if it cannot
occur in a vertex minimal graph G with r/e(G) > k. Note that any fcreducible
configuration is also (k + l)reducible. When applying the discharging method in
Theorem 5.19, we require the following known lemma, which is a simple application
of Eulers Formula:
Proposition 5.9 Given a planar graph G,
E ('(/) 4) + E ) 4> = 8
feF(G) vev(G)
Proof: Let G be a planar graph. By Eulers Formula (Proposition 4.1), we have
that
2 = \V(G)\ \E(G)\ + \F(G)\.
Multiplying by 4 gives
8 = 2\E{G)\ 4E(G) + 2\E{G)\ 4F(G).
Applying Propositions 4.4 and 4.3 and grouping terms gives the desired result.
We also require a large independent set, which is given from the following theorem:
69
Theorem 5.10 (Steinberg, Tovey [86]) Every planar trianglefree graph on n ver
tices has an independent set of size at least .
O
The main tool we use to determine when configurations are fcreducible is the
Combinatorial Nullstellensatz, which is applied to certain graph configurations.
5.2 Reducible Configurations
In the lemmas in this section, we let k G N and introduce fcreducible configura
tions that will be used to prove our main result. Let Â£ : V(G) > 2R be a function on
V(G) such that Â£(u) = k for each v G V(G). Thus, Â£{v) denotes a list of k available
labels for v. In each proof, we take G to be a vertex minimal graph with r/e(G) > k.
Then, we define a proper subgraph G' of G with V(G') C V(G). By the choice of G,
G' has a lucky labeling l such that Â£(v) G Â£{v) for all v G V(G'). This labeling of G'
is then extended to a lucky labeling of G by defining Â£(v) for v G V(G) V{G'). We
discuss the details of this approach in Lemmas 5.11 and 5.12. The remaining lemmas
are similar in approach, so we include fewer details in the proofs.
Lemma 5.11 The following configurations are k^reducible in the class of graphs with
girth at least 5:
(a) a vertex v with d(u) < k, and
ueN(v)
(b) a vertex v with r neighbors of degree 1 and a set Q of 2neighbors {v\,... ,vq}
each having a (k 1)~ neighbor other than v, say v[,... ,v'q, respectively, such
that v[,... ,v'q are independent and 1 + r(k 1) + (k d(v'f) 1) > d(v).
ViÂ£Q
Proof: Assume G is a vertex minimal graph with rjg(G) > k containing the
configuration described in (a). Let G' = G {v}. Since G is vertex minimal, r)e(G') <
k. Let l be a lucky labeling from Â£ on V(G'). Our aim is to choose Â£(v) from Â£{v) to
extend the lucky labeling of G' to a lucky labeling of G. Note that the only unsatisfied
edges of G are those incident to neighbors of v. Let e be an edge incident to a neighbor
70
u of v. If e = uv, then e is satisfied when Â£(v) Y Yf, Yw) ~ Sg'(u). If e = uw
weN(v)
for some w ^ v, then e is satisfied when Â£(v) Y Sg'(w) Sg'(u). Thus, picking Â£{v)
distinct from at most Y1 d(u) values ensures that all edges of G are satisfied. Since
ueN(v)
Y^ d(u) < k, there exists Â£(v) in C{v) that can be used to extend the lucky labeling
ueN(v)
of G' to a lucky labeling of G. Therefore, rje(G) < k, a contradiction.
Now, assume G is a vertex minimal graph with girth at least 5 and rje(G) > k
containing the configuration described in (b). Let R be the set of r 1neighbors of
v. Let G' = G (HU Q). Since girth(G) > 5, Q is independent. Therefore, for
each i E {1,... ,q}, there are at least \C{vi)\ dciv') choices for Â£(vi) that ensure
all edges incident to v\ are satisfied in G. Consider vw in E(G). If w E V(G'), vw
is satished when Y, ^(x) Y Sg'(w) Sg'(v) Also, if w E R, then vw is satisfied
xeRUQ
when Y, x) Y Yv) ~ SgYv) If w = Vi for some Vi E Q, then vw is satished when
xeRUQ
Y1 Yx) Y Â£(v) + Â£(w) SgYv) Therefore, we must avoid at most d(v) values
xeRUQ
for Y1 h(cc) in order to satisfy all edges incident to v. Recall that each vertex
xeRUQ
in R and Q have k and k d(u') labels, respectively, that avoid restricted sums.
Proposition 5.7 guarantees at least 1 + r(k 1) + Y1 (k d(v[) 1) available values
vieQ'
for Y, h(u>)+ Y1 Yw) Since, by assumption, l + r{k 1)+ Y1 (k~d(v'j) l) > d(v),
wÂ£zR wÂ£Q V{(zQ
there is at least one choice for Â£{w) for each w in RUQ that completes a lucky labeling
of G. Thus, rje(G) < k, a contradiction.
Lemma 5.12 The following configurations are 8reducible in the class of graphs of
girth at least 6:
(a) a 6vertex v having six 2neighbors one of which has a 3 neighbor, and
(b) a 7vertex v having seven 2neighbors two of which have 4 neighbors.
Proof: Let G be a vertex minimal graph of girth at least 6 with rje(G) > 8.
To the contrary, suppose G contains the configuration described in (a). Let u be a
71
2neighbor of v having a 3_neighbor. Let G' = G {u, v}. Let Â£ : V(G') > E be a
lucky labeling of G' such that Â£(v) G C{v) for each v G V(G).
The only unsatished edges of G are those incident to neighbors of u or v. To
satisfy the unsatished edges not incident to n or r, we avoid at most two values from
Â£(u) and at most five values from Â£(v). Note that Â£(tt) > 8 and Â£(u) > 8. Thus,
there are at least six labels available for u and at least three available for v. To satisfy
the edges incident to m or r, f() Â£(v) must avoid at most seven values. Corollary
5.8 gives at least eight values for Â£{u) Â£{v) from available labels. Thus, there are
labels that complete a lucky labeling of G. Hence, rje(G) < 8, a contradiction.
V
Figure 5.1: An 8reducible configuration.
Now, we prove part (b). To the contrary suppose G contains the configuration
described in (b). Let u\,... ,uj be the 2neighbors of v whose other neighbors are
u[,... ,u'7, respectively, where u[ and v!2 are 4_vertices. Since the most restrictions
on labels occurs when d{u= d{v!2) = 4, we assume this is the case. Let N(u[)
{i} = {w\,W2,w3} and N{v!2) {m2} = {w^w^w^} (see Figure 5.1). Consider
G' = G {v,Ui,U2} Let Â£ : V(G') > E be a lucky labeling of G' such that
Â£{v) G C{v) for each v G V(G). The only unsatished edges of G are those incident
to u[, v!2, and neighbors of v. The following function has factors that correspond to
unsatished edges, where x, y, and z represent the possible values of Â£(v), Â£(u\), and
72
l(v,2), respectively:
f(x,y,z)
7 / 7
11 [v + z + ^2Â£(ui) ~x~Â£(u'i)
i=1 V 3=3
3 3
7
Yl(x + Â£(u[) Sc'iu'i))
i=3
Y[(y + Sg'K) S'G'(w*)) n( + Sg'^u'x> ~ sG'{^i))
i=1 i=l
(x + Â£(u[) y Sa'iy^i)) (x + Â£(i4) z Sg<(u'2)).
The coefficient of x7y6z7 in f(x, y, z) is equal to its coefficient in (y + z x)7x5y3z3(x
y)(x z), which is 490. The Combinatorial Nullstellensatz (Theorem 4.7) implies that
there is a choice of labels for Â£(v), Â£(u\), and Â£(U2) from lists of size at least 8 that
make / nonzero. Thus, these labels induce a lucky labeling of G. Hence, rje(G) < 8,
a contradiction.
Lemma 5.13 A configuration that is an induced cycle with vertices V1V2V3V4V5 such
that d(v 1) < 17, dfvf) = d(v5) = 2, d(v3) < 7, and d(v4) <7 is 19reducible.
Figure 5.2: A reducible configuration.
Proof: Let G be a vertex minimal graph with r)t(G) > 19. Suppose to the
contrary that G contains the configuration in Figure 5.2. Since the most restrictions
on labels occurs when d(v 1) = 17 and d(v3) = d(v4) = 7, we assume this is the
case. Let G' = G {^2,^5}. Let Â£ : V{G') > E be a lucky labeling of G' such that
Â£{v) G C{v) for each v G V(G). The unsatished edges are those incident to v3,... ,v5.
The following function has factors corresponding to the unsatished edges where X2
73
and X5 represent labels of V2 and V5, respectively:
f(x2,x5) =(SG'(v 1) + x2 + x5 Â£(vi) Â£(v3)) (Â£(vi) + Â£{v3) x2 SGfiv3))
(x2 + SG>(v3) x5 SGfiv4)) (x5 + SGfiv4) Â£{vfi) Â£{v4))
(Â£(vi) + Â£{v4) x2x5 SG'(v 1))
(SGfiw) SG/(v4) x5)
weNG,(v 4){^3}
Yl (SG'(w) SG'(vi)x2x5)
wÂ£Nq/(v 1)
(SGfiw) SG/(v3) x2).
weNG,(v 3){^4}
The coefficient of x}fix\A in f(x2, Â£5) is the same as xfifx\ in {x2 + X5)u(x2 X5),
which is (j7) (g7). The Combinatorial Nullstellensatz (Theorem 4.7) implies that
rjfiG) < 19, a contradiction.
Lemma 5.14 Let P(t2,..., t_i) be the path v4 vn such that for each i E {2,..., n
1} the vertex ry has L 1neighbors and d(vf) = 2 \~U. The configurations P( 1,0,1),
P( 1,1,1), P( 1,1, 0, 0), P(0,1,0, 0), P( 1, 0, 0, 0), and P(0, 0, 0, 0, 0) are 3reducible.
mv7
'Vi V2 Vs V4 V5
mv6 mv7 *V8
I**
'Vi V2 Vs V4 V5
mv7 mvs
^2 vs V4 V5 vq
(a)
pv7
(b)
^ ><
Tl 1^2 ^3 ^4 ^5 ^6 Vi V2 Vs V4 V5 Vff
(V7
(c)
Vi V2 Vs V4 V5 Vq Vf
(d)
(e)
(f)
Figure 5.3: Some 3reducible configurations.
Proof: For the entirety of this proof, let G be a vertex minimal graph with
Ve(G) > 3.
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Suppose to the contrary that G contains P(1,0,1), see Figure 5.3a. Let G' =
G {U3, ve, v7}. Let Â£ : V(G1) > E be a lucky labeling of G' such that Â£(v) G C(v) for
each v G V(G). The unsatished edges are those incident to v2,V3,v4. The following
function has factors corresponding to the unsatished edges where X3, xe, and x7
represent labels of v3, v6, and v7, respectively:
f(x3,x6,x7) = (SG>(v 1) Â£(vi) X3 x6) {Â£{v\) +X3 + X3 Â£{v2))
(Â£(vi) + x3 + x& Â£{v2) Â£{v4)) (Â£(v2) + Â£(v4) SG>(v4) x3 x7)
(Â£(v5) + x3 + x7 Â£(v4)) (Â£(v5) + x3 + x7 SG'(v5)).
The coefhcient of x\x^Xj in / is 9. The Combinatorial Nullstellensatz (Theorem
4.7) implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(1,0,1) is 3reducible.
Suppose to the contrary that G contains P(1,1,1), see Figure 5.3b. Let G' =
G {v3,ve,v7, ug}. Let Â£ : V(G') GRbea lucky labeling of G' such that Â£(v) G C{v)
for each v G V(G). The unsatished edges are those incident to v2,V3,v4. The following
function has factors corresponding to the unsatished edges where X3, xe, x7, and x$
represent labels of v3, v6, v7, and v8, respectively:
f(x3,x6,x7,x8) = (SG>(v 1) Â£(vi) X3 x6) {Â£{v\) +X3 + X3 Â£{v2))
(Â£(vi) +X3 + X3 Â£{v2) x7 Â£{v4)) (Â£(v2) + Â£{v4) + x7 x3)
(Â£(v2) + Â£(v4) + x7 Â£(v5) x3 x8) (Â£(v5) + x3 + x8 Â£{v4))
(Â£(v5) + x3 + x8 SG>(vr_;))
The coefhcient of x1xgx7x1 is 15. The Combinatorial Nullstellensatz (Theorem 4.7)
implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(1,0,1) is 3reducible.
Suppose G contains P(l, 1,0,0), see Figure 5.3c. Let G' = G {v3,v4,v7,v8}. Let
Â£ : V(G') > E be a lucky labeling of G' such that Â£(v) G C(v) for each v G V(G). The
75
unsatisfied edges are those incident to v2, ,v3. The following function has factors
corresponding to the unsatisfied edges where X3, x4, x7, and x$ represent labels of v3,
v4, Vj, and vs, respectively:
f(x3,X4,X7,Xg) = (x3 + x7 + Â£(v4) SG'(v 1)) (x3 + x7 + Â£(v4) Â£(v2))
(x4 + x8 + Â£(v2) x3 x7 Â£(vi)) (x4 + x8 + Â£{v2)  x3)
(x4 + x8 + d(u2) x3  Â£(v5)) (x3 + Â£(v5) x4 Â£(v6))
(x4 + Â£(v6) SG'(v6))
The coefficient of X3x\x^x\ is 8. The Combinatorial Nullstellensatz (Theorem 4.7)
implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(l, 1,0,0) is 3reducible.
Suppose G contains P(0,1,0,0), see Figure 5.3d. Let G' = G {v3,v4,v7}. Let
Â£ : V(G') > E be a lucky labeling of G' such that Â£(v) E C(v) for each v E V(G). The
unsatisfied edges are those incident to v2,... ,v5. The following function has factors
corresponding to the unsatisfied edges where x3, x4, and x7 represent labels of v3, v4,
and v7, respectively:
f(x3,x4,x7) = (SG'(v 1) Â£(vi) x3) (Â£(vi) +x3 Â£(v2) x7 x4)
(Â£(v2) + x7 + x4 x3) {Â£{v2) + x7 + x4 x3 Â£{v5))
(x3 + Â£(v5) x4 Â£(v6)) (x4 + Â£(v6) SGr(v6)).
The coefficient of x^x^Xj is 6. The Combinatorial Nullstellensatz (Theorem 4.7)
implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(0,1,0,0) is 3reducible.
Suppose G contains P(l,0,0,0), see Figure 5.3e. Let G' = G {v3,v4,v7}. Let
Â£ : V(G') > E be a lucky labeling of G' such that Â£(v) E C(v) for each v E V(G). The
unsatisfied edges are those incident to v2,... ,v5. The following function has factors
corresponding to the unsatisfied edges where x3, x4, and x7 represent labels of V3, v4,
76
and v7, respectively:
f(x3,x4,x7) = (SG'(vi) Â£(vi) x3 x7) (Â£(v2) Â£{vf) x3 x7)
(Â£(v2) +x4 Â£{vf) x3 x7) (Â£(v2) + x4 x3 Â£(v5))
(Â£(v6) + x4 x3 Â£(v5)) (Â£(v6) +x4 SGr(v6)).
The coefficient of x\x\x7 is 7. The Combinatorial Nullstellensatz (Theorem 4.7)
implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(l, 0,0,0) is 3reducible.
Suppose G contains P(0,0, 0, 0, 0), see Figure 5.3f. Let G' = G {v3, v^, v3}. Let
Â£ : V(G') > E be a lucky labeling of G' such that Â£(v) E C(v) for each v E V(G). The
unsatisfied edges are those incident to v2,... ,v3. The following function has factors
corresponding to the unsatisfied edges where x3, x, and x3 represent labels of v3, v,
and v5, respectively:
f(x3,x4,x5) = (SGr(vi) Â£(vi) x3) (Â£(vi) +x3 Â£{v2) x4) (Â£(v2) + x4 x3 x5)
(x3 + x5 x4 Â£(v6)) (x4 + Â£(v6) x5 Â£(v7)) (x5 + Â£(v7) SG>(v7)).
The coefficient of x^x^x^ is 7. The Combinatorial Nullstellensatz (Theorem 4.7)
implies that Â£ can be extended to a lucky labeling of G, a contradiction. Thus,
P(0,0,0,0,0) is 3reducible.
5.3 Lucky Labeling Results
In this section, we prove the four results enumerated in Theorem 5.6.
Theorem 5.15 If G is a planar graph with girth(G) > 5, then r)t(G) < 19.
Proof: Let G be a planar graph with girth at least 5 and suppose that G is
vertex minimal with rje(G) > 19. By Proposition 5.3, mad(G) < 10/3. Assign each
vertex v an initial charge d(v), and apply the following discharging rules:
(Rl) each 1vertex receives 7/3 charge from its neighbor;
77
(R2) each 2vertex
(a) with two 8+neighbors receives 2/3 charge from each neighbor,
(b) with a 4_neighbor and a 15+neighbor receives 4/3 charge from its 15+
neighbor, or
(c) with a 10+neighbor and a neighbor of degree 5, 6, or 7 receives 1 charge
from its 10+neighbor and 1/3 charge from its other neighbor; and
(R3) each 3vertex receives 1/3 charge from a 6+neighbor.
A contradiction with mad(G) < 10/3 occurs if the discharging rules reallocate
charge so that every vertex has final charge at least 10/3; we show that this is the
case.
By Lemma 5.11 (a), each 1vertex has a 19+neighbor, 2vertices have neighbors
with degree sum at least 19, and 3vertices have at least one 6+neighbor. Thus,
by the discharging rules, 3_vertices have final charge 10/3. Since 4vertices neither
give nor receive charge, they have final charge 4.
Vertices of degree d with d E {5,6,7} give charge when incident to 3_vertices.
By the discharging rules, they give away at most d/3 charge. This results in a final
charge of at least d  = y > y, since d > 5.
We consider vertices of degree d with d E {8,9}. By Lemma 5.11 (a), each
9vertex has at least one 3+neighbor. Also, each 8vertex has at least two 3+
neighbors or at least one 4+neighbor. By the discharging rules, the final charge of
any 9vertex is at least 9 8   = y and the final charge of any 8vertex is at
least min{8 6  2 , 8 7 Â§} = y.
Next, we consider vertices of degree d with d E {10,11}. By Lemma 5.11 (b),
these vertices have no 2neighbors with a 7_neighbor. Thus, these vertices have
final charge at least (i y =  > y, since d > 10.
78
Here, we consider vertices of degree d with d E {12,13,14}. Let v be such a vertex.
By Lemma 5.11 (b), v has no 2neighbor with a 4_neighbor. By Lemma 5.11 (b)
and Lemma 5.13, v has at most two 2neighbors each having a 7_neighbor. By the
discharging rules, v has final charge at least d 2(1) (d 2) () = yA > y, since
d > 12.
Now, we consider vertices of degree d with d E {15,16,17}. As before, Lem
mas 5.11 (b) and 5.13 guarantee that such vertices have at most one 2neighbor with
7_neighbors. Thus, these vertices give at most 1() + (d 1)  charge. Hence, they
have final charge at least yA > y, since d > 15.
Finally, consider an 18+vertex v of degree d. Let r be the number of 1neighbors
of v. Let U = {ui,U2, uq} be the set of 2neighbors of v. For each Ui let N(ui)
{u} = {'} Let T = {u'i E U : dfu'f) < 7} and let \T\ = t. Since G[T] is planar
with girth at least 5, Theorem 5.10 guarantees at least yy vertices in T that form an
independent set. By Lemma 5.11 (b), d > 18r + ll(yy) + 1. Thus,
11 14 , ,
d > 18r + t + . (5.1)
O O
The hnal charge of v is at least d (h r t). Hence, v has final charge
at least  r 11. From (5.1),  r > yr + + y When r > 1 or t > 4,
the hnal charge is at least y. When r = 0 and t < 3, the vertex v has hnal charge at
least d (h t) > yh > y, since d > 18.
Theorem 5.16 If G is a planar graph with girth(G) > 6, then rje(G) < 9.
Proof: Let G be a planar graph with girth at least 6 and suppose G is vertex
minimal with %(G) > 9. By Proposition 5.3, mad(G) < 3. Assign each vertex v an
initial charge of d{v) and apply the following discharging rules:
(Rl) each 1vertex receives 2 charges from its neighbor; and
(R2) each 2vertex
79
(a) with one 8+neighbor and one 5 neighbor receives 1 charge from its 8+
neighbor,
(b) with one 7+neighbor and one 4_neighbor receives 1 charge from its 7+
neighbor,
(c) with one 6+neighbor and one 3_neighbor receives 1 charge from its 6+
neighbor, or
(d) receives 1/2 charge from each neighbor, otherwise.
A contradiction with mad(G) < 3 occurs if the discharging rules reallocate charge so
that every vertex has final charge at least 3; we show this is the case.
By Lemma 5.11 (a), each 1vertex has a 9+neighbor and each 2vertex has
neighbors with degree sum at least 9. Under the discharging rules, 1vertices and 2
vertices gain charge 2 and 1, respectively, and 3vertices neither gain nor lose charge.
Thus, 3_vertices have final charge 3.
By Lemma 5.11 (b), each 4vertex v has no 1neighbor and has at most one
2neighbor whose other neighbor is a 6_vertex. Therefore, each 4vertex has final
charge at least 4 Similarly, each 5vertex has no 1neighbor and has at most four
2neighbors having another 7_neighbor. Therefore, each 5vertex has final charge
at least 5 4(A), as desired.
If v is a 6vertex, then Lemma 5.11 implies that v has no 1neighbor. Moreover,
Lemma 5.12 implies that if v has six 2neighbors then at most one of them has a
3_neighbor. Hence, v has charge at least 6 max{l + 4(),6()}, which is 3, as
desired.
Similarly by Lemma 5.11, a 7vertex v has no 1neighbor. Moreover, by Lemma
5.12, if v has seven 2neighbors, at most one of them has a 4_neighbor. Thus, v has
charge at least 7 max{2 + 4(), 1 + 6(), 7()}, which is at least 3 as desired.
80
Finally, if v is a hvertex with d> 8, then by Lemma 5.11 (b) we have
d>8r + 3q + l, (5.2)
where r is the number of 1neighbors and q is the number of 2neighbors having a
5_neighbor. The final charge on v is at least d 2r q \ {d r q) =  Â§r \q.
By (5.2), v has final charge at least (8r + 3q + 1) r \q = r + q + When
r > 1 or q > 3, this final charge is at least 3. Otherwise, when r = 0 and q < 2, v
has final charge at least d 2 \{d 2) = > 3, since d > 8.
Theorem 5.17 If G is a planar graph with girth(G) > 7, then rje(G) < 8.
Proof: Let G be a planar graph with girth at least 7 and suppose G is a vertex
minimal planar graph with r/e(G) > 9. By Proposition 5.3, mad(G) < 14/5. Assign
each vertex v an initial charge of d{v) and apply the following discharging rules:
(Rl) each 1vertex receives 9/5 charge from its neighbor; and
(R2) each 2vertex
(a) with one 3_neighbor and one 6+neighbor receives 4/5 charge from its
6+neighbor,
(b) with one 3neighbor and one 5neighbor receives 1/5 and 3/5 charge, re
spectively,
(c) with two 4neighbors receives 2/5 charge from each neighbor,
(d) with one 4neighbor and one 5+neighbor receives 1/5 and 3/5 charge,
respectively, or
(e) with two 5+neighbors receives 2/5 charge from each neighbor.
A contradiction with mad(G) < 14/5 occurs if the discharging rules reallocate charge
so that every vertex has final charge at least 14/5; we show this is the case.
81
By Lemma 5.11 (a), each 1vertex has an 8+neighbor and each 2vertex has
neighbors with degree sum at least 8. Under the discharging rules, 1vertices and 2
vertices gain 9/5 and 4/5 charge, respectively. If v is a 3vertex, then Lemma 5.11 (b)
implies that v has at most one 2neighbor with a 5neighbor other than v. Thus, v
gives at most 1/5 charge. Hence, 3_vertices have final charge at least 14/5.
If v is a 4vertex, then Lemma 5.11 (b) implies that v has at most one 2neighbor
with a 4_neighbor other than v. Thus, v has final charge at least 4 1 () 3 () = 3.
If v is a 5vertex, then Lemma 5.11 (b) implies that v has at most one 2neighbor
with a 4_neighbor. Thus, v has final charge at least 5 1 () 4 () >
If v is a 6vertex, then Lemma 5.11 (b) implies that v has at most one 2neighbor
with a 3_neighbor, and at most one 2neighbor having a 4_neighbor. Thus, v has
final charge at least 6 1 1 () 4 = 3.
If v is a 7vertex, then Lemma 5.11 (b) implies that v has at most one 2neighbor
with a 3_neighbor, and has at most two 2neighbors with a 4_neighbor. Thus, v
has final charge at least 7 1 () 2 () 4 () = //.
If v is an 8vertex, then Lemma 5.11 (b) implies that v has at most one 1
neighbor, at most two 2neighbors with a 3_neighbor, and at most two 2neighbors
with a 4_neighbor. Moreover, if v has a 1neighbor, then v does not have a 2
neighbor with a 3_neighbor. Since the discharging rules allocate charge to neighbors
with these constraints, v has final charge at least 8max l + 7 () 2 + 6 () }
If v is a hvertex with d > 9, then Lemma 5.11 (b) implies that v has at most
 1neighbors, at most  neighbors that are either a 1vertex or a 2vertex with a
a 4 neighbor. Since v gives more charge to neighbors of low degree, we assume
v has as many low degree neighbors as possible. Hence, v has final charge at least
17
5 '
3 neighbor, and at most
d
3
neighbors that are either a 1vertex or a 2vertex with
d d
4 8
d d
3 4
jwjd, which is at least 3 since d >
82
9. Therefore, all vertices have final charge at least 14/5 and we obtain a contradiction.
We call a hvertex lonely if it is in exactly one face of G. We say that a nonlonely
3+vertex v is unique to a face / of G if it is incident to a cutedge uv such that
d(u) > 1 and uv is also in /.
Lemma 5.18 Let f be a face in a planar graph G with ec cutedges such that f has
s lonely vertices, and t 3+ vertices unique to f. We have s +  < ec.
Proof: We apply induction on ec. If ec = 0, then s = t = 0 and the inequality
holds. In the following two cases, given some face / containing a cutedge uv, let G'
be the graph obtained by contracting the edge uv to a vertex w. Let f be the face
in G' corresponding to /. Let s' and t' be the number of lonely vertices in f and the
number of 3+vertices unique to f, respectively.
Case 1: u or v is lonely.
Without loss of generality, assume u is lonely. If v is also lonely, then w is lonely
and therefore s' = s 1. If v is not lonely, then w is not lonely and still s' = s 1.
Vertices unique to / are not affected by the contraction, giving t! = t. Since f has
ec 1 cutedges, the induction hypothesis implies that s' +  < ec 1. Therefore,
s +  < ec.
Case 2: u and v are unique to /.
Since u and v are not lonely, w is not lonely and s' = s. After contracting uv, either
w is unique to / and t! = t 1 or w is not unique to / and t' = t 2, which yields
t' + 1 < t < t' + 2. By the induction hypothesis, s' + ^ < ec 1. Since t < t' + 2, we
have s +  < ec, as desired.
Theorem 5.19 If G is a planar graph with girth(G) > 26, then %(G) < 3.
Proof: Let G be planar with girth at least 26 and suppose G is vertex minimal with
r]e(G) > 3. Assign each vertex v an initial charge d{v), each face / an initial charge
83
1(f), and apply the following discharging rules:
(Rl) each 1vertex receives 2 charges from its incident face and 1 charge from its
neighbor;
(R2) each 2vertex receives 2 charges from its incident face if it is lonely, otherwise
it receives 1 from each incident face;
(R3) each 3vertex with a 1neighbor and
(a) incident to two faces receives 1 charge from each incident face, or
(b) incident to one face receives 2 charges from its face;
(R4) each 3vertex without a 1neighbor and
(a) incident to three faces receives  charge from each incident face,
(b) incident to two faces receives \ charge from each incident face, or
(c) incident to one face receives 1 charge from its face;
(R5) each 4vertex that has a 1neighbor and is
(a) incident to three faces receives  charge from each incident face, or
(b) lonely or unique to some face / receives 1 charge from /; and
(R6) each 5vertex that has two 1neighbors and is
(a) incident to three faces receives  charge from each incident face, or
(b) lonely or unique to some face / receives 1 charge from /.
A contradiction with Proposition 5.9 occurs if the discharging rules reallocate charge
so that every vertex and face has charge at least 4; we show this is the case.
Lemma 5.11 (a) implies that a 1vertex has a 3+neighbor. Lemma 5.11 (b)
implies that a 4_vertex has at most one 1neighbor, a 5vertex has at most two
84
1neighbors, and in general a hvertex has at most neighbors of degree 1. Since
vertices only give charge to 1neighbors, 6+vertices have final charge at least 4. Note
that if v is a hvertex with d E {3,4,5}, at most d 3 neighbors of degree 1, and
incident to at most two faces, then v is unique to a face. Thus, all vertices have final
charge at least 4 under the discharging rules.
We turn our attention to the final charge of faces. By Theorem 5.5 and the choice
of G, G is connected and each face contains at least one cycle. Therefore, each face
has length at least 26. Let Rf be the set of vertices incident to a face / that are either
a 2vertex or a 3vertex that is not lonely and has one 1neighbor. Let / be a face
with s lonely vertices, t unique vertices, and r vertices in Rf. Lemma 5.18 implies
that / has at least s +  cut edges. Thus,
1(f) > 26 + 2s + i. (5.3)
A combination of the reducible configurations in Lemma 5.14 implies that there are
at most four consecutive vertices from Rf in any cycle of /. Thus,
r <
)(/(/)2si)
By the discharging rules, / has final charge at least
1(f) 2 str (i(/) 2 str) = h(f)   r.
(5.4)
By (5.4),
2 4222 422
1(f)s1r > 1(f)s1
3u; 3 3 3 ~ 3 J 3 3 3
Therefore, the final charge of / is at least
(1(f) 2st)
Â§/) 2 *>) = 2 *),
(5.5)
which is at least 4 when 1(f) 2s 5 > 30. When 1(f) E {26,
final charge at least 4.
29}, (5.5) gives
85
Since [25] shows cr(Cn) = x(C'ra) for n > 3 and cr(G) < 77(G), we have q{C2n+i) > 3
for n > 13. By Theorem 5.19, we have the following immediate corollary:
Corollary 5.20 If n > 13, then qfXC2n+ i) = 3.
86
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COMPUTATIONALAPPROACHESINGRAPHTHEORY by AXELTHOMASBRANDT B.S.,Mathematics,OhioNorthernUniversity,2010 M.S.,Mathematics,MiamiUniversity,2012 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy AppliedMathematics 2016
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ThisthesisfortheDoctorofPhilosophydegreeby AxelThomasBrandt hasbeenapprovedforthe DepartmentofMathematicalandStatisticalSciences by FlorianPfender,Advisor MichaelFerrara,Chair PaulHorn MichaelJacobson SogolJahanbekam April28,2016 ii
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Brandt,AxelThomasPh.D.,AppliedMathematics ComputationalApproachesinGraphTheory ThesisdirectedbyAssociateProfessorFlorianPfender ABSTRACT Weconsiderthreeproblemsingraphtheoryandintertwineadvancedmathematicaltheorywithsophisticatedcomputationalmethodstoobtainsolutions.The rsttwoproblemsutilizethetheoryofagalgebrasincombinationwithsemidenite programmingtoobtainresultsinextremalgraphtheory.ThethirdproblemusescomputationaltoolstofacilitateanovelapplicationoftheCombinatorialNullstellensatz withinproofsusingthedischargingmethod. First,weinvestigatetheminimumedgedensitybetweenpartsofafourpartite graphthatguaranteestheexistenceofatriangle.Ourapproachreliesheavilyupon developingaagalgebramodelthatviewstheproblemascoloringtheedgesofa completegraph. Next,wepursueaconstructionthatobtainsthemaximumnumberofinduced k cyclesinagraphofxedorder.In1975,PippengerandGolumbicconjecturedthat thebalancediteratedblowupofacycleistheextremalconstructionforallinduced cyclesonatleast5vertices.Thisconjecturewasarmedearlierthisyearforinduced 5cyclesongraphsofspecicsizes.Usingasimilarapproach,ourresultsarmthis conjectureforinduced6cyclesongraphsoforder 6 m for m 2 N andprovideasecond pieceofevidencethattheconjecturemaybetrue.Initialcomputationalexperiments indicatethatwemaybeabletoextendourmethodstoarmthisconjecturefor k =7 ; 8 ongraphsofcertainorder. Finally,weseektodeterminethe"luckynumber"ofgraphs.Foranyvertex labeling ` : V G R wesaythat ` is lucky iftheneighborhoodsforeverypairof iii
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adjacentverticeshavedistinctsumsofvertexlabels.Theleastinteger k forwhicha graphhasaluckylabelingusinglabelsfrom f 1 ;:::;k g isits luckynumber .In2009, Czerwiski,Grytczuk,andelaznyconjecturedthattheluckynumberofagraphis atmostitschromaticnumber.Thisconjecturehasonlybeenconrmedforafew graphclasses.Ourresultsprovidenewandimprovedboundsformultiplefamiliesof planargraphs.Indoingso,weconrmthisconjecturefornonbipartiteplanargraphs withgirthatleast26. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:FlorianPfender iv
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Formyfuturestudents v
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ACKNOWLEDGMENT Firstandforemost,thankyoutomywifeKimberlyforyourlove,understanding, andsupportovertheyearsandparticularlythroughoutmygraduatecareer;without itIwouldnotbewhoIamtoday. IwouldliketoexpressanimmenseamountofgratitudetoFlo,myadvisor,for hisguidancethroughmydoctoralstudies,andamassiveamountofappreciationto Mike,mycommitteechairandseeminglyunocialadvisor,forhisactivementorship inmygrowthasbothamathematicianandaneducator.Fortheirextensiveeorts inmydevelopmentIamdeeplyindebtedandinexplicablythankful. ToBrent,myocemate,thankyouforputtingupwithmeonadailybasisfor thepastfouryears.ToJenny,Cathy,Tim,andPhil,formergraduatestudents,thank youforyourencouragementandexample.ToHencandEric,alsoformergraduate students,thankyouforpushingmetolearnL A T E XandTikzearlyoninmygraduate studies.ToDevon,afellowgraduatestudent,thankyouforyourpatiencewithmy manynaiveoptimizationquestions. Thankyoutomycoauthorsforcollaboratingwithme,andparticularlytoFlo, Bernard,Sogol,andJennyforcollaborationsthatledtoresultscontainedinthis manuscript.Additionally,thankyoutothemanyAmericantaxpayerswithoutwhom thefollowingNSFgrantsthatpartiallyfundedmyresearchwouldnothaveexisted: DGE0742434,DMS1427526,DMS1500662,andIUSE1539692. Furthermore,thankyoutothemanyteacherswhohavefosteredbothmylove ofmathematicsandpassionforeducationovertheyears:Dan,Mihai,Sandy,Don, Kristen,Laura,Dave,andJohn.Inparticular,ahugethankyoutoTao,mymaster's advisor,forencouragingmetopursuedoctoralstudies. Finally,thankyoutomyfamilyandfriendsfortheirsupportandencouragement overtheyears.Inparticular,thankyoutoBillandJanforconvincingmetopursue doctoralstudiesandthankyousomuchtomom,Bruce,andEllieforproofreading thisdocument. vi
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TABLEOFCONTENTS LISTOFFIGURES................................viii COMPUTATIONSINEXTREMALGRAPHTHEORY............1 1.AnIntroductiontoExtremalGraphTheory..............2 1.1StabilityMethod.........................4 1.2FlagAlgebras...........................5 1.3ConvexOptimization.......................9 1.4ObtainingExtremalExamples.................11 2.DensityBoundsinMultipartiteGraphs................16 2.1TrianglesinMultipartiteGraphs................16 2.2AProposedExtremalConstruction...............17 2.3AFlagAlgebraModel......................19 2.4MultipartiteResult........................20 3.InducedCycles..............................28 3.1Induced5Cycles.........................30 3.2Induced6Cycles.........................32 3.2.1TopLevelStructureoftheExtremalGraph......33 3.2.2IterativeStructureoftheExtremalGraph.......53 COMPUTATIONSINGRAPHCOLORING...................56 4.AnIntroductiontoGraphColoring..................57 4.1TheFourColorTheoremandtheDischargingMethod....58 4.2ListColoringandtheCombinatorialNullstellensatz......62 5.LuckyLabeling.............................65 5.1NotationandTools........................68 5.2ReducibleCongurations....................70 5.3LuckyLabelingResults.....................77 REFERENCES...................................87 vii
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LISTOFFIGURES Figure 2.1Thegeneralstructureofaconjecturedextremalgraphfor d 3 4 .......17 2.2Flagalgebraconstraintsforvertexlabeledgraphmodelinga4partite, K 3 freegraph.................................19 2.3Comparingthenumberofagsonsmallvertexsets............20 2.4Flagalgebraconstraintsfor3edgecoloredcompletegraphmodelinga balanced,4partite, K 3 freegraph......................21 2.5Flagalgebraconstraintsfor4edgecoloredcompletegraphmodelinga 12partite, K 3 freegraph...........................27 3.1Progressionofobtainingabalanced,iteratedblowupof C 4 ........29 3.2Theextremalgraphfor i C 6 C k 6 ......................32 3.3Graphsin C 22 and C 3 .............................33 3.4Theembeddingsofaninduced P 5 withonefunkyedgein G [ X 0 ;X 1 ] ...40 3.5Theembeddingsofaninduced P 5 withfunkyedgein G [ X 0 ;X 1 ] ......41 3.6Theembeddingsofaninduced P 5 withfunkyedgein G [ X 0 ;X 3 ] ......43 3.7Induced C 6 swithallfunkyedgesincidenttoasinglevertex........46 5.1An8reducibleconguration.........................72 5.2Areducibleconguration...........................73 5.3Some3reduciblecongurations.......................74 viii
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COMPUTATIONSINEXTREMALGRAPHTHEORY Extremalgraphtheoryisabroadareaofresearchingraphtheory.Atitscore,this areaofresearchseekstoidentifyandlearnaboutthresholdphenomena.Theoriginatingprobleminthisareaconcernsmaximizingthenumberofedgesinagraphthat avoidsaspeciedforbiddensubgraphandidentifyingthegraphorgraphsthatobtain thismaximum.InjointworkwithFlorianPfenderandBernardLidick,thetheory ofagalgebrasisusedtowardidentifyingthresholdsandtheassociatedgraphsfor twoproblems.Theauthorcontributedtotherstproblembyidentifyingagalgebra constraintsthatmodeltherstproblemandwritingcodefortheirimplementation. Theauthorcontributedtothesecondproblembyidentifyingagsthatwerecrucial inobtainingusefulcomputationalbounds,andverifyingmanyofthedetailswithin stabilityarguments. 1
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CHAPTER1 ANINTRODUCTIONTOEXTREMALGRAPHTHEORY Extremalgraphtheoryfocusesonboththerelationsbetweenvariousgraphparameters,suchasorder,size,minimumandmaximumdegree,andchromaticnumber, andalsothevaluesoftheseparametersthatguaranteecertaingraphproperties.More formally,givenaproperty P andaparameter forafamily F ofgraphs,weseek todeterminetheminimumormaximumvalue m forwhicheverygraph G 2F with G >m or G m impliesthat G containsacycle.Theminimumsuch m is n ,andthefamilyoftreeson n vertices constitutestheextremalgraphsforthisproblem.Inthiscase, F isthefamilyofall graphson n vertices, P isthepropertythatagraphcontainsacycle, G represents j E G j ,and m = n Thistoyexamplecanalsobeviewedassearchingforthemaximum m suchthat everygraph G oforder n with j E G j
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Countingthenumberofcopiesofaxedgraph H inahostgraph G isanatural extensionoftheproblemofdeterminingif H isasubgraphof G .Forinstance,Erds andStone[38]guaranteethatalargenumberofcliquesappearinsucientlylarge graphswitharelativelysmallnumberofedgesbeyondwhatTurn'stheoremrequires toguaranteeasingleclique.ErdsandSimonovits[37]usetheaforementionedresult toprovidetheproportionofedgesinextremalgraphsoforder n as n !1 dependent solelyuponthemaximumchromaticnumberofagraphintheforbiddenfamily. Formuchofthischapterwedirectourattentiontolargegraphs.Inthissetting, ratherthanexamining ex n; F forlarge n itisperhapsnaturaltoexaminethe proportionofthetotalnumberofpossibleedgesinanextremalgraph.Thefollowingis astraightforwardexampleofhowthisperspectivecanbemoreusefulthancomparing theexactedgecountsofvariouslargegraphs: Proposition1.1 Foreveryfamily F ex n; F n 2 isdecreasingas n !1 Proof: Let H beanextremal F freegraphon m vertices.Let n m and G 1 ;:::;G t bethe )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(m n inducedsubgraphsof H on n vertices.Noticethateachedgeof H isin )]TJ/F21 7.9701 Tf 5.479 4.378 Td [(m )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 n )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 ofthe G i s.Thus, ex m; F m )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 n )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 = j E H j m )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 t X i =1 j E G i j ex n; F m n : Since )]TJ/F21 7.9701 Tf 5.48 4.378 Td [(m n = )]TJ/F21 7.9701 Tf 5.479 4.378 Td [(m )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 n )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 = m m )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 n n )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 = )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(m 2 = )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n 2 ,wehave ex m; F )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(m 2 ex n; F )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n 2 : Goingforwardinthischapter,werstdiscussthestructuralpropertiesofgraphs havingaparameterclosetothatofanextremalgraph.Then,wediscussavery recentlyintroducedtoolinextremalcombinatorics,thetheoryofagalgebras,thatwe willusetoobtainextremalresultsinsubsequentchapters.Asimplementationofthe 3
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agalgebramethodreliesoncomputationaltools,theconclusionofthischapterwill provideanoverviewoftheoptimizationtheoryunderlyingagalgebracomputations andanexamplethereof. Beforeproceeding,itwillbeusefultointroducenotationthatdescribesthe asymptoticbehavioroffunctions.Let n beanintegervariablethattendstoinnityandlet n beapositivefunctionand f n beanarbitraryfunction.Itis saidthat f 2 O providedthat j f j 0 ,thereexists > 0 and n suchthatif G is F freeandif,for n>n j E G j >ex n; F )]TJ/F20 11.9552 Tf 11.955 0 Td [(n 2 ; then G canbeobtainedfrom T n;p bychangingatmost n 2 edges. Inthecontextofmaximizingedgesintheforbiddensubgraphproblem,thistheoremstatesthatan F freegraphwithnearly ex n;F edgesisstructurallysimilarto T n; F )]TJ/F15 11.9552 Tf 11.32 0 Td [(1 .Bystructurallysimilar,wemeanthatthenearlyextremalgraphcan bealteredtoobtain T n; F )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 bychangingrelativelyfewedges. Asthenamesuggests,therearemorestabilitytheorems;seethebookbyBollobs [14]orthesurveybySimonovits[83].Manystabilityresultsareutilizedasaprecursor 4
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toprovinggraphstobeextremal.Ourresultsintheupcomingchapterswillusethis approach,whichbeginswithagraph H whoseparameter foraproperty P exceeds thatoftheproposedextremalgraph G .Weupperbound H by G + forsome small > 0 andthenusetheclosenessof H and G toguaranteethataspects of H 'sstructurealignwithaspectsof G 'sstructure. 1.2FlagAlgebras In2007,Razborov[75]presentedaformalmodelcalledagalgebrasinwhicha numberofcentralproblemsofextremalgraphtheorycanbeexpressed.Specically, thismodelallowstheformalizationofseveralimportanttoolsinextremalcombinatoricssuchastheCauchySchwarzinequalitytobeappliedtoproblemsingraph theory.Insomesense,agalgebrasprovideamethodthatenablesndingthebest possiblewayinwhichtoapplytheCauchySchwarzinequalitytoacollectionofgraph parameters.Thisendeavoriscompletedbywayofsolvingasemideniteprogram, whichisconvenientgiventhenumberofpubliclyavailable,optimized,semidenite programsolvers. Whilewewilloverviewtheprocessofsolvingsuchaprograminthenextsection, theremainderofthissectionwillbedevotedtoprovidingaconceptualunderstanding ofagalgebras.Althoughwewilldealprimarilywithsimplegraphsinthecontext ofsubgraphdensities,agalgebrascanalsobeusedtorepresentorientedgraphs [42,52,59,60,61,78]andhypergraphs[9,10,43,70,77].Forclarity,wewill usepicturesofgraphsratherthantheirnamesasoftenaspossible.Foramore detailedpresentationofthetheoryofagalgebras,wedirectthereadertothepaper ofRazborov[76],forwhichhewasawardedtheRobbinsPrizefromtheAmerican MathematicalSocietyin2013,andthedissertationofSperfeld[85]fromwhichwe borrowsomeusefulexamples. Todeneags,weneedrstthenotionofa type .If isthesubgraphinduced byalllabeledverticesofagraph G ,then G hastype andiscalleda ag.Note 5
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thatunlabeledgraphsaresaidtohavetype0.Focusingonlabeledgraphsallowsus todistinguishbetweenisomorphiccopiesofagraph,anendeavorthatiscrucialboth intheoryandalsoincomputationalimplementation,whereallgraphsarelabeled. Wewillwidelyviewa agasitsdensityinanarbitrarilylargegraph.This densityisformallyobtainedfromprobabilities,whichwedescribehere.Acollectionof sets V 1 ;:::;V t isa sunowerwithcenter C if V i V j = C foreverydistinct i;j 2 [ t ] .For acollectionof ags F 1 ;:::;F t ,let G beagraphwithenoughverticestocontainthe sunower F 1 ;:::;F t withcenter V ,thatis j V G j P t i =1 j V F i j)78(j V j t )]TJ/F15 11.9552 Tf 10.231 0 Td [(1 Choosein V G uniformlyatrandomasunower V 1 ;:::;V t withcenter V and j V i j = j V F i j forall i 2 [ t ] .Wedenoteby p F 1 ;:::;F t ; G theprobabilitythat G j V i isisomorphicto F i forall i 2 [ t ] .Fortwoexamplesoftype0withasingle F i p ; G istheprobabilityofpickingtwoadjacentverticesin G ,and p ; G istheprobabilityofpickingthreepairwiseadjacentverticesin G Forapotentiallymoreinsightfulexample,consider p v ; v ; G .Here, = f v g andeach F i hasoneadditionalvertex.Thus,weareconcernedwiththe probabilityofpickingtwoverticesof V G nf v g withoneadjacentto v andtheother notadjacentto v .Therearetwopossiblesunowers,specically v and v Enumeratingthepossiblesunowersisakeypartofagmultiplication. Beforedeningagmultiplication,werstdescribeaagalgerba chainrule that allowsforthederivationofnumerousequalitiesbetweenprobabilities.Foragraph G andacollectionof ags F 1 ;:::;F t ,let ` j V G j sothat,for s 2 [ t ] ` islarge enoughsothatagraphon ` verticescouldcontainthesunower F 1 ;:::;F s with center V and G hasenoughverticestocontainthesunower H;F s +1 ;:::;F t with center V ,where H isa agon ` vertices.Thenwecanrewrite p F 1 ;:::;F t ; G as asumoftheproduct p F 1 ;:::;F s ; H p F s +1 ;:::;F t ; G overall ags H on ` vertices. Thinkingofthisasastatementinvolvingconditionalprobabilities,wepresentthe followingexampleregardingtheedgedensityofagraphinwhichthesunowers 6
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F 1 ;:::;F s = and H;F s +1 ;:::;F t = H where H rangesoverallunlabeled graphson ` =3 vertices: p ; G = p ; p ; G + p ; p ; G + p ; p ; G + p ; p ; G : Astheprobabilityofedgesappearinginxedgraphson3verticesiseasytocalculate, wecanwritetheedgedensityofagraphasalinearcombinationoftheprobabilities ofgraphson3verticesappearingin G asfollows: p ; G = 3 3 p ; G + 2 3 p ; G + 1 3 p ; G + 0 3 p ; G : .1 Ifweabusenotationtointerpretthepictureofa ag F as p F ; F 0 fora largeenough ag F 0 ,thenwecanstateasymptoticversionsofnotableresults.For example,considerGoodman's1959result[47]ontheminimumnumberoftriangles inagraphanditscomplement. Theorem1.3Goodman[47] Let G beagraphon n vertices.Then G andits complementcontainatleast 8 > > > > > > < > > > > > > : 1 3 m m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ; for n =2 m; 1 3 2 m m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m +1 ; for n =4 m +1 ; and 1 3 2 m m +1 m )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ; for n =4 m +3 ; triangles. Asymptotically,thisstatesthat G anditscomplementmustcontainatleast 1 4 n 3 + o triangles.Thiscanbestatedinthelanguageofagalgebrasas + 1 4 wheretheadditionofagsisthenaturaladditionofrealnumbers.Mantel'sTheorem [69]on ex n;K 3 canbestatedasymptoticallyasmaximizing subjectto =0 inordertoconclude 1 2 7
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Usingthechainrule,wecanformallydenethemultiplicationoftwo ags F 1 and F 2 .Let ` belargeenoughsothatagraphon ` verticescouldcontainthesunower F 1 ;F 2 withcenter V .Then,denethemultiplicationofags F 1 F 2 tobethe sumoftheproduct p F 1 ;F 2 ; F p F ; G overall ags F on ` vertices.Towritethe productoftwoagsasalinearcombination,weuseallagsuptoisomorphismthat canbeobtainedbyaddingedgestothesunower F 1 ;F 2 withcenter V between V F 1 n V and V F 2 n V .Forexample,thesquareoftheedgedensitycanbe expressedusingthesubgraphdensitiesofgraphsonasfollows: 2 = = 1 3 + 1 3 + 1 3 + 2 3 + 2 3 + ; wheredottedlinesrepresentthepotentialofedgestocompletethesunowerand whereweabusenotationtoallowthepictureofagraph H torepresent p H ; G Because isa0agon2vertices,multiplyingitbyitselfgivesnooverlapinthe resultingsunower.Thus,weexamineall0agson4vertices.Theweightsoneach probabilityaccountforthenumberofwaysthatamatchingappearsfromaselection oftwovertexpairs. Asanotherexample,considerthefollowingproduct: v v = v = 1 2 v + 1 2 v + 1 2 v : Because v and v are1agson2vertices,theirproductisa1agon3vertices containinganoriented P 3 with asitsinteriorvertex.Asbefore,weconsiderall possiblewaystocompletethissunower.Weweighteachtermby 1 2 toaccountfor the2waystoobtainanoriented P 3 withinteriorvertex whenpickingverticesto buildthesunower. Underthissunowerdenitionofagmultiplication,wehavetheabilitytowrite anyquadratictermofagsasalinearcombinationofagsonmorevertices.This processisacriticalstepinestablishingaagalgebramodelthatcanbesolved computationally. 8
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Ascombinatorialargumentsfrequentlyuseaveragingarguments,wedescribethe averagingoperator,whichwillenableustoprovideaversionoftheCauchySchwarz Inequalityinthelanguageofagalgebras.Let F bea agand G denote F with nolabels.Dene q F tobetheprobabilitythataninducedembeddingof in G givesa agisomorphicto F .The averagingoperator fora ag F ,denoted J F K isdenedtobe q F G .When =0 ,weomitthesubscriptandsimplywrite J F K Viewedroughly,theaveragingoperatordeletesthelabelsonaagandmultipliesit byanappropriateprobability.Withthisnotation,theCauchySchwarzinequality canbestatedasfollows: Theorem1.4FlagAlgebraicCauchySchwarzInequality[75] Let f;g belinearcombinationsof ags.Then J f 2 K J g 2 K J fg K 2 Theinequality J f 2 K 0 istrueasanimmediateconsequenceofTheorem1.4by setting g =1 .Weturnourattentiontowardestablishingabroadunderstandingof thecomputationalmethodsusedinconjunctionwithagalgebras. 1.3ConvexOptimization Convexoptimizationisparticularlyfriendlybecauseanylocalmaximumorminimumisalsoaglobalmaximumorminimum.Semideniteprogrammingisa subeldofconvexoptimizationthatwerelyuponforourresults.Inwhatfollowswe discusssemideniteprogramminginaagalgebracontext.Wedirectthereaderto abookbyHelmberg[57]foramorethoroughtreatmentofsemideniteprogramming forcombinatorialoptimization. Ourprimarygoalistoverifyaninequalityoftheform f d where f isalinear combinationof agsand d 2 R .Ultimately,weaimtomaximize d sothatthe boundon f isasgoodaspossible.ForaagalgebrastatementofTheorem1.3, + correspondsto f and 1 4 correspondstothemaximumpossible d Towardbroadlydiscussingsemideniteprogrammingasusedwithagalgebras, let F ` denotethespaceofall agson ` verticesuptoisomorphism,andlet f be 9
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alinearcombinationof ags.Asusedinlinearalgebra,let S n denotethesetof allsymmetric, n n realmatrices.Inthiscontext,wedeneamatrix A 2 S n tobe positivesemidenite ,denoted A 0 ,if x T A x 0 forall x 2 F ` n .Equivalently, A 0 ifeveryeigenvalueof A isnonnegative. AsaconsequenceofTheorem1.4,wehavethefollowingcorollaryforpositive semidenite A 2 S n and x 2 F ` n : J x T A x K 0 : Thus,ifthereexistspositivesemidenite A 2 S n sothat f )]TJ/F20 11.9552 Tf 11.956 0 Td [(d J x T A x K ; thenwecanconcludethat f d .Oftentimes,suchaswithMantel'sTheorem, additionalknownordesiredaginequalitiesi.e.constraintsareeithernecessaryor helpfulinsolvingtheproblemathand.If C 2 R m n and c 2 R m 1 sothat CA + c representsacollectionof m agalgebraconstraints,thenourgoalistosolvethe followingsemideniteprogram: SDP 8 > > > > > > < > > > > > > : maximize d subjectto C x + c d; A 0 : Severalexamplesofthisproblemformationcanbefoundin[8],[49],[54],[55],and [58]. Thereareseveralbenetstoencodingagalgebraproblemsassemideniteprograms.First,semideniteprogramsareclosedunderduals,whichmeansthedual programisalsosemidenite.Morespecically,strongdualityofsemideniteprograms guaranteesthatifasolutiontoasemideniteprogramexists,thenthecorresponding dualprogramhasanidenticalsolution.Thisallowsforprimaldualmethodsthat 10
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alternatesolvingtheprimalanddualoptimizationproblemstowardthecommonsolutiontobothproblems.Thereareavarietyofsolvers,includingCSDP[20]and SDPA[45],forsemideniteprogramsthatrunveryeciently.Addingtotheconvenienceofsemideniteprogramsolvers,Vaughan[91]hasdevelopedafreesoftware calledFlagmaticwithwhichtoencodeagalgebrasasasemideniteprogram. Althoughwewillnotdiscussitinourresults,wewouldberemissnottomentiontheconcernofsmallerrorsresultingfromoatingpointnumericalcomputations. Ideally,wewouldliketosimplyrounddecimalentriesofthesolver'ssolutionmatrixtorationalnumbersandmaintainafeasiblepositivesemidenitematrix.This isgenerallypossiblewhentheeigenvaluesofthesolutionmatrixareallsuciently largepositivenumbers.However,ifthesolutionmatrixhaseigenvaluesclosetozero, thenroundingentriesriskscreatingnegativeeigenvalues,inwhichcasewenolonger haveasolution.Ifweassumeinsteadthateigenvaluesclosetozeroareinfactzero inanoptimalsolution,thenatechniqueinBaber'sdissertation[7]ishelpful.Unfortunately,thereisnouniversalsolutiontondingsharpresultsusingthismethod. Manyresultsarefoundsimplybyexaminingconstraintsandmakingsmallchanges toturnarounded,nonfeasiblesolutionintoafeasiblesolution. 1.4ObtainingExtremalExamples Inthissection,weoverviewthestandardapplicationofagalgebrasindeterminingextremalconstructionsandprovideanexampleofitsusethroughtwoproofsof Mantel'sTheorem. Thegeneralapproachtypicallybeginswithaknownlowerorupperboundobtainedfromaconstruction.Presumably,thisboundistheconjecturedextremalvalue. Aftertranslatingtheextremalproblemunderconsiderationintothelanguageofag algebras,asemideniteprogramisusedtoobtainanupperorlowerboundcloseto theknownbound.Asmentionedintheprevioussection,theresultingsolutionmatrix isroundedtoobtainafeasiblesolutionmatrixthatisprovablypositivesemidenite. 11
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Fromthisroundedsolution,subgraphsthatappearwith0densitycanbeidentied. Usingtheseaddedstructuralpropertiesofforbiddensubgraphs,stabilityarguments areusedtoobtaintheextremalconstruction. Itisperhapsworthmentioningthat,inpractice,roundingthesolutionmatrix iscommonlythelaststepundertaken.Ratherthaninitiallyroundingtheentire solutionmatrix,theresultingupperorlowerboundisitselfroundedandutilized forstabilityargumentsinanattempttodetermineifitiscloseenoughforthose stabilityargumentstohold.Ifstabilityargumentsholdfortheroundedbound,then theprocessisreversedtodeterminewhatboundwouldbenecessaryforthestability argumentstoremaintrue.Thesolutionmatrixisthenroundedwiththegoalof obtainingaboundsucientforthestabilityargumentstogivethedesiredresult.In thiscontext,usingecientsemideniteprogramsolverstoobtainprecisenumerical boundsisinvaluable.Usingpowerfulcomputationalmachinesallowscomputations tobeperformedonlargerags,whichgeneratemoreinequalitiesandcontributeto obtainingprecisenumericalbounds. TheremainderofthissectionisdedicatedtopresentingtwoproofsofMantel's Theoremanddeterminingtheextremalconstructionusingagalgebras. Theorem1.5Mantel[69] Atrianglefreegraphon n verticescontainsatmost n 2 4 + o edges. Proof: Atheoreticalapproach. Notethat.1canbewrittenas = + 2 3 + 1 3 : .2 Foratrianglefreegraph,weassume =0 : .3 12
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Combining.2and.3gives = 2 3 + 1 3 : .4 Since 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 v 2 0 ,wehavefrom1.3and.4that 0 t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 v 2  s 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 v +4 v +4 v { + o = s 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 v +4 v { + o =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 + 4 3 + o =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 + o .5 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 + o ; fromwhichweconclude 1 2 + o .Interpreting asaprobability,wehavethat thenumberofedgesinatrianglefreegraphisatmost 1 2 )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n 2 = n 2 4 + o ,asdesired. TowarddeterminingtheextremalconstructionforMantel'sTheorem,assume =0 and = 1 2 .Undertheseassumptions,.5gives 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 + o = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2 3 + o : Since 0 byitsnatureasaprobability,theaboveinequalityimplies =0 Thus and aretheonlyunlabeledagson3verticesthatappearinthe extremalconstruction.Itisstraightforwardtoarguethatconstructinganextremal graphwiththisrestrictionon3vertexsubgraphsgivesabalanced,bipartitegraph. Proof: Acomputationalapproach. Notefromthepreviousapproachthatitsucestoshow 1 2 + o .Our intentistond c 1 ;c 2 ;c 3 2 R sothat 0 c 1 + c 2 + c 3 + o 13
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foreverygraph.Aftersummingthisinequalitywith.4,wehave c 1 + 1 3 + c 2 + 2 3 + c 3 + o : .6 Noticethatforourdesiredresultweneed c 1 1 2 c 2 1 6 ,and c 3 )]TJ/F18 7.9701 Tf 23.113 4.707 Td [(1 6 Let A = ac cb beapositivesemidenitematrixover R .Thenfor x = v ; v ,wehave xAx T 0 .Thus, 0 u v v ; v 0 @ ac cb 1 A v ; v T } ~ = s a v + v + b v + v +2 c v + v { = a + a +2 c 3 + b +2 c 3 + b + o = a + a +2 c 3 + b +2 c 3 + o : Setting c 1 = a c 2 = a +2 c 3 ,and c 3 = b +2 c 3 ,then.6implies a + 1+ a +2 c 3 + 2+ b +2 c 3 + o max a; 1+ a +2 c 3 ; 2+ b +2 c 3 + + + o =max a; 1+ a +2 c 3 ; 2+ b +2 c 3 + o ; .7 wherethelastlinefollowsfrom 1= + + since =0 .Wecan representtheaboveinequalityasthesemideniteprogram SDP 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : Minimize d subjectto a d; 1+ a +2 c 3 d; 2+ b +2 c 3 d; ac cb 0 : SolvingSDPwithacomputationalsolvergivesasolutionof )]TJ/F18 7.9701 Tf 10.765 5.032 Td [(0 : 5 )]TJ/F18 7.9701 Tf 6.587 0 Td [(0 : 5 )]TJ/F18 7.9701 Tf 6.586 0 Td [(0 : 50 : 5 .Thus,.7 implies max 1 2 ; 1 6 ; 1 2 + o = 1 2 + o ,asdesired. 14
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Towarddeterminingtheextremalconstruction,assume = 1 2 .Thesolutionof SDPand.6givethefollowinginequalities: 1 2 = 1 2 + 1 6 + 1 2 + o ; 1 + 1 3 + + o : Since =0 ,wehave 1= + + .Subtractingthisfromtheabove inequalitygives 0 )]TJ/F15 11.9552 Tf 23.113 8.088 Td [(2 3 + o ; whichimplies =0 ,asbefore.Determiningtheextremalconstructionnishes asbefore. 15
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CHAPTER2 DENSITYBOUNDSINMULTIPARTITEGRAPHS Inthischapterweareinterestedprimarilyinmultipartitegraphs.Agraphis k partite ifitsverticescanbepartitionedinto k independentsets,whicharereferred toas parts .A k partitegraphis complete ifthereisanedgebetweeneverypairof verticesfromdistinctparts.Turngraphsarespecialcasesofcompletemultipartite graphsinwhichthenumberofverticesineachpartdiersbyatmostone. ConsideringtheexistencecriteriafortrianglesingraphsestablishedinMantel's Theorem[69]andtheworktowardcomputationallyidentifyingtrianglesingraphs [4,26,62],itisperhapsnaturaltopursueotherexistencecriteriafortrianglesin specicgraphclasses.Thegoalofthischapteristousethetheoryofagalgebrasto identifyaconditionthatguaranteestheexistenceofatrianglein4partitegraphs. 2.1TrianglesinMultipartiteGraphs WhileMantel'sTheoremandTurn'sTheoremaretypicallyintroducedascountingthresholds,theycanalsobeviewedasedgedensitythresholds.Forexample, Mantel'sTheoremcanbestatedinthefollowingform: Theorem2.1Mantel[69] Everybipartitegraph G with j E G j > 1 2 )]TJ/F24 7.9701 Tf 5.48 4.379 Td [(j V G j 2 containsa K 3 Inabipartitegraph,alloftheedgedensityoccursbetweenthetwopartsofthe partition.Inmultipartitegraphswithmorethantwoparts,itisnaturaltowonderif thereisanedgedensitythresholdbetweenpartsthatalsoguaranteesa K 3 or,more generally,a K k forsome k 2 N Tothisend,let G bean ` partitegraphwithpartition V 1 ;V 2 ;:::;V ` .Forbrevity, dene j G j := j V G j and jj G jj := j E G j .For i 6 = j ,dene d ij tobetheedgedensity between V i and V j ,thatis d ij := G [ V i [ V j ] j V i jj V j j : 16
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Let d k ` denotetheminimumdensitybetweenpartsin G thatguaranteesthat G containsa K k ,thatis d k ` :=min f c 2 R :min i;j d ij >c K k G g : MotivatedbyaquestionofErds[35],in[18]Bondy,Shen,Thomass,and Thomassendetermined d 3 3 = ,thegoldenratio.Theyalsoshowedthat d 3 existsandequals 1 2 andthat d 3 4 0 : 51 .Theyspeculatedthat d 3 ` > 1 2 forallnite ` However,Pfender[72]proved d 3 ` = 1 2 for ` 12 .Pfenderalsoprovedthat d k ` = k )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 forlargeenough ` Ourworkdeterminesanupperboundfor d 3 4 .Thebulkofthiseortliesinformulatingconstraintssothatagalgebracomputationsareimplementedonabalanced, 4partitegraph. 2.2AProposedExtremalConstruction GeneralizingalemmafromBondy,Shen,Thomass,andThomassen[18],Sperfeld [84]providesaconstructionthatlowerbounds d 3 4 .Thisconstructionisablowupof a4partitegraphwithatmost3verticesineachpart.Aswehavenotspecically mentioneditpreviously,the blowup ofagraph G isobtainedbyreplacingeachvertex withasetofindependentverticesandaddingedgesbetweenverticesiftheyarein setsthatcorrespondtoadjacentverticesin G .ThegeneralstructureofSperfeld's constructionisshowninFigure2.1.Fortheskepticalreader,wedescribethegraph's Figure2.1:Thegeneralstructureofaconjecturedextremalgraphfor d 3 4 underlyingstructureandincludeaproofthatitis K 3 free,whichimpliesthatany blowupisalso K 3 free. 17
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Proposition2.2 Let G beagraphwithvertexset x i ;y i ;z i for i 2 Z 4 suchthat G [ x 0 ;x 1 ;x 2 ;x 3 ]= C 4 G [ x i ;y i ;z i ;y i +2 ;z i +2 ]= K 3 ; 2 withbipartitiondenotedbyindices, and y 0 y 1 ;y 2 y 3 ;z 0 z 1 ;z 2 z 3 2 E G .Wehavethat G is K 3 free. Proof: Sinceeach f x i ;y i ;z i : i 2 Z 4 g isanindependentset,anytrianglemust witnessdistinctindices.Since N x i = f x i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ;x i +1 ;y i +2 ;z i +2 g ,ifatrianglecontains x i ,thenitmustalsocontain x i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 or x i +1 .However,thisisnotpossiblebecause N x i N x i +1 = ; .Thus,atriangledoesnotcontainany x i Since G [ y 0 ;:::;y 3 ]= G [ z 0 ;:::;z 3 ]=2 K 2 ,atrianglemustcontainatleastone y i andatleastone z j .Bysymmetry,itsucestoconsiderifatrianglecancontain y 0 and y 1 .Thisisnotpossiblebecause N y 0 = f y 1 ;x 2 ;y 2 ;z 2 g and N y 1 = f y 0 ;x 3 ;y 3 ;z 3 g Therefore, G is K 3 free. Usingheuristicstooptimizethesizesofeachsetintheblowupextremalconstruction,wegetthefollowinglowerboundfor d 3 4 : d 3 4 0 : 511342 : .1 Fortheinterestedreader,weconsideraverticalsymmetryinSperfeld'sconstruction [84]thatheprovesoptimizesthelowerboundgivenbythegeneralstructurepresented inFigure2.1.Weview x i ;y i ;z i for i 2 Z 4 asindependentsetsofverticesandassume j x 0 j = j x 1 j j y 0 j = j y 1 j j z 0 j = j z 1 j j x 2 j = j x 3 j j y 2 j = j y 3 j ,and j z 2 j = j z 3 j andthenuse Mathematicatonumericallymaximizetheminimumof d ij for i 6 = j 2 Z 4 .Specically, thisistheminimumoftheset f ax + zc;x 2 + y 2 ;a 2 + b 2 ; 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(ax g whenviewing x;y;z astherespectiveproportionsof x i ;y i ;z i in j x i j + j y i j + j z i j for i 2f 0 ; 1 g and a;b;c astherespectiveproportionsof x i ;y i ;z i in j x i j + j y i j + j z i j for i 2f 2 ; 3 g 18
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2.3AFlagAlgebraModel Inthissection,weconsiderarstapproachtomodelinga K 3 free4partitegraph usingagalgebras.Sincea4partitegraphisfrequentlyviewedasproperly4vertexcolored,itisnaturaltoconsiderlabelingverticestospecifypartitesets.Forinstance, labelingverticesofagsfrom Z 4 canbeviewedasspecifyinginwhichpartiteseta vertexresides. Inthisframework,establishingconstraintsforaagalgebramodelbecomesvery straightforward.Forinstance,ensuringnonedgesbetweenverticesofthesamelabel donotoccuriseasilyaccomplishedbysettingeachedgedensitybetweenverticesof thesamelabeltobe0.Toensurethattrianglesdonotappear,wewouldalsoset thedensityoftheremaining4possiblepropervertexlabelingsfrom Z 4 to K 3 equal to0.Inconjunctionwiththeprevious4constraintsonedgedensities,thissuces toguaranteeouragalgebraconstraintsmodela4partite, K 3 freegraph.These constraintsarepicturedinFigure2.2. 0 0 = 1 1 = 2 2 = 3 3 = 0 = = = = 0 0 0 2 2 2 3 3 3 1 1 1 Figure2.2:Flagalgebraconstraintsforvertexlabeledgraphmodelinga4partite, K 3 freegraph. Whileaddingconstraintstothesemideniteprogramoftenimprovesthesolver's computationtime,thesizeofthesemideniteprogramistheprimaryfactorimpacting computationtime.Withthesizeofthesemideniteprogramdependentuponthesize andtypeofagsused,thenumberofagsonaxednumberofverticesisareasonable indicatorofthesemideniteprogram'ssize.Inlargepartthisisbecausevariablesfor thesemideniteprogramaregeneratedbymultiplyingpairsofags,asseeninthe computationalproofofTheorem1.5. Consideringthatthereare16agson2vertices,72agson3vertices,and407 19
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#Vertices 2 3 4 5 Vertex4ColorModel 16 72 407 2235 Edge3ColorModel 3 8 28 137 Figure2.3:Comparingthenumberofagsonsmallvertexsets. agson4verticessatisfyingtheconstraintsinFigure2.2,thenumberofagsgrows veryquicklyastheirsizeincreases.Thus,computationsforthisvertexlabeledmodel arerestrictedtousingverysmallagsthatgiveweakerbounds.Figure2.3compares thenumberofagsonsmallnumbersofverticesinthismodelwiththoseforthe modelpresentedinthenextsection.Thecomputationallimitationsresultingfrom thelargenumberofsmallagsinthemodelpresentedinthissectionserveasour primarymotivationforusinganalternativemodel. 2.4MultipartiteResult WeproceedundertheassumptionthatSperfeld'sconstructionisextremalfor d 3 4 .Webeginbydescribingsomeadditionalassumptionswecanmake.Then,we discusshowagalgebrascanbeusedtoencodeboththeknownstructureofand ourassumptionsaboutanextremalgraph.Finally,weuseourmodeltoobtain computationalboundsfor d 3 4 Insteadofcoloringverticestodenotepartitesets,weseektopartitionverticesby 3edgecoloringacompletegraphunderspecicconstraints.Specically,weseekto encodewhetherverticesareinthesamepartitesetusingred,blue,andgreenedges. Tothisend,wewillviewrededgesascorrespondingtoedgesina4partitegraph, blueedgesascorrespondingtononedgesbetweenverticesindistinctpartitesetsof a4partitegraph,andgreenedgesascorrespondingtononedgesbetweenverticesin thesamepartitesetofa4partitegraph. Asolidedge, ,willrepresentarededgeinthe3edgecoloredcompletegraph. Adottededge, ,willrepresentablueedgeinthe3edgecoloredcompletegraph. 20
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Adashededge, ,willrepresentagreenedgeinthe3edgecoloredcompletegraph. Amultiedgewillrepresentanyofthecorrespondingcolorededges.Forexample, denotesthepossibilityofeitheraredorblueedgeinthe3edgecoloredcomplete graph. Weclaimthatwecanassumeourgraphisbalancedintwoways.First,wecan assumethatthepartitesetsarebalanced;otherwisewecanblowupeverypartby anappropriateconstanttobalancethesets.Notethatthisscalingofentireparts preservesthedensitiesbetweenparts.Also,wecanassumethatedgedensitybetween partsisbalanced,thatis d ij = d k` forall 1 i
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models,aspresentedinFigure2.3,motivatesourchoicetomodelthisproblemasa 3edgecoloredcompletegraph. Lemma2.3 Let G bea3edgecoloredcompletegraphusingcolorsred,blue,and green.Let H beagraphobtainedfrom G bydeletingblueandgreenedges.If G satises =0 ; =0 ; =0 ; = 1 4 ; and P 1 2 3 4 1 2 3 4 = P 1 2 3 4 1 2 3 4 ; then H isa K 3 free,balanced,4partitegraphwith d ij = d k` forall 1 i
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BytheCauchySchwarzInequalitywehave n 2 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o n 2 = n 2 n 2 2 X i 2 Z 4 x i 2 2 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(n 2 = n 2 8 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o n 2 as n !1 .Thus, 1 4 : Theaddedconstraintof = 1 4 forcesequalityintheCauchySchwarzInequality, whichoccurspreciselywhen x i = x j for i;j 2 Z 4 .Hence,thepartsof H arebalanced. Itremainstoshowthatedgedensitiesbetweenpartitesetsin H arebalanced. Tothisend,let i;j;k;` 2 Z 4 suchthat i 6 = j and k 6 = ` .Observethat d ij d k` + o where o as n !1 ,istheprobabilityofindependentlypickingtwopairsofvertices v i ;v j 2 V i ;V j and v k ;v ` 2 V k ;V ` sothat v i v j and v k v ` arerededges.Fordistinct i;j;k;` 2 Z 4 ,theproducts d 2 ij + o d ij d ik + o ,and d ij d k` + o canberepresented astherespectiveconditionalprobabilities P 1 2 3 4 1 2 3 4 ;P 1 2 3 4 1 2 3 4 ; and P 1 2 3 4 1 2 3 4 : .2 Thus,iftheconditionalprobabilitiesin.2areequal,then d 2 ij + o = d ij d ik + o = d ij d k` + o : Underthisassumption, d ij = d k` as n !1 forall i 6 = j 2 Z 4 and k 6 = ` 2 Z 4 ,whichgivesthedesiredbalance.Inordertoimplementthissucientequality withinagalgebracomputationswiththeotherconstraints,wedesireanunlabeled representationofthisequality.Inwhatfollows,weexaminehowtoestablish d 2 ij + o = d ij d ik + o ,whichissucienttoensure d ij = d k` forall i 6 = j 2 Z 4 and k 6 = ` 2 Z 4 Weworkbackwardsfromthedesiredequalityandassume d 2 ij + o = d ij d ik + o : .3 23
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First,weexamine d 2 ij + o = P 1 2 3 4 1 2 3 4 .Noticethatthereare )]TJ/F18 7.9701 Tf 5.48 4.379 Td [(4 2 ways topick i 6 = j 2 Z 4 with2subsequentchoicesfromwhichparttopickeachpair.Then, thereare )]TJ/F21 7.9701 Tf 6.675 4.977 Td [(n 4 2 + o waystopicktwoverticesfromeachpart.Thus, 1 2 3 4 = 4 2 2 n 4 4 + o = 3 n 4 64 + o : .4 Similarly,for d ij d ik + o = P 1 2 3 4 1 2 3 4 thereare 4! waystopickdistinct i;j;k 2 Z 4 )]TJ/F21 7.9701 Tf 6.675 4.977 Td [(n 4 2 + o waystopick2verticesin V i ,and n 4 verticestopickinboth V j and V k .Thus, 1 2 3 4 =4! n 4 3 = 6 n 4 64 + o : .5 Combining.4and.5in.3givesthefollowingequalities: d 2 ij + o = d ij d ik + o ; P 1 2 3 4 1 2 3 4 = P 1 2 3 4 1 2 3 4 ; 64 3 n 4 1 2 3 4 + o = 64 6 n 4 1 2 3 4 + o ; 2 1 2 3 4 = 1 2 3 4 : .6 Weusetheaveragingoperatortorepresentthelabeledagsin.6aslinearcombinationsofunlabeledags. Listingallofthepossibilitiesforthelefthandsideof.6gives 1 2 3 4 = 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 : .7 Fortheaveragingofeachlabeledag,wecounthowmanyofthe 4! vertexlabelingsof anunlabeledagresultinthecorrespondinglabeledag.Sinceallverticesmustbe incidenttoonegreendottedandtworedsolidedgesin ,thereare4choicesfor label1thatallforcethechoiceforlabel2,whichinturngives2waystoappropriately label3and4.Since1mustbeincidenttoallthreeedgecolorsin ,bothofthe2 24
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choicesforlabel1forcethechoiceoftheremaininglabels.Since2mustbeincident toallthreeedgecolorsin ,bothofthe2choicesforlabel2forcethechoiceofthe remaininglabels.Sinceallverticesmustbeincidenttoallthreeedgecolorsin all4choicesforlabel1forcethechoiceoftheremaininglabels.Therefore,from.7 wehave t 1 2 3 4  = 4 2 4! + 4 4! + 4 4! + 4 4! : .8 Listingallofthepossibilitiesfortherighthandsideof.6gives 1 2 3 4 = 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 + 1 2 3 4 : .9 Since 1 2 3 4 1 2 3 4 ,and 1 2 3 4 eachcontaina ,therstconstraintgives = = =0 .Sincethereisonlyonegreendashededgein ,there are2waystolabel1and2,bothofwhichforcethelabelsof3and4.Sincethereis onlyonegreendashededgein ,thereare2waystolabel1and2,afterwhich thereare2waystolabel3and4.Since1mustbeincidenttoallthreeedgecolors in ,thereisonly1choiceforlabel1,whichforcestheremaininglabels.Since2 isincidenttoallthreeedgecolorsin ,thereisonly1choiceforlabel2,which forcestheremaininglabels.Sincethereisonlyonegreendashededgein ,there are2waystolabel1and2,bothofwhichforcethelabelsof3and4.Therefore,from .9wehave t 1 2 3 4  = 2 4! + 2 2 4! + 1 4! + 1 4! + 2 4! : .10 Simplifyingthecombinationof.6,.8,and.10gives 8 +4 +4 = +2 + + ; whichisthedesiredunlabeledconstraintthatforcesthedesiredequalityin.3. 25
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AsaresultofthisLemma,wehaveconstraintsinthelanguageofagalgebras whoseadditiontoagalgebracomputationsgiveboundsforthedesiredgraph.Using theseconstraints,wecanruncomputationstondanupperboundfor d 3 4 Theorem2.4 d 3 4 < 0 : 513243 : Proof: Thisfollowsfromastandardapplicationoftheplainagalgebramethod. Wemaximize withtheconstraintslistedinLemma2.3.Thecomputationswere performedon7verticesandasolutiontothesemideniteprogramwasfoundusing CSDP. Forcomparisonwiththepreviouslymentionedlowerbound.1,wehavethe followingregarding d 3 4 : 0 : 511342 d 3 4 < 0 : 513243 : Theobservantreadermayhavenoticedthatthis3edgecoloredmodeldoesnot fullymodelSperfeld's[84]construction,whichispicturedinFigure2.1.Inparticular, the3edgecoloredmodeldoesnotenforcetherenedpartitioninto12partswiththe appropriateedgesbetweenparts.Below,wediscussawayinwhichtoaddanedge color,sayorange,torenethe3edgecolormodeltowardSperfeld'sextremalexample. Inthis4edgecoloredmodel,redandblueedgesretaintheirsamemeaning,and greenandorangetogetherprovidethe3partstructuretoeachpartiteset.Specically, orangeedgesareusedtoreneeachpartinthe4partitionbyrecoloringgreenedges toindicatewhenverticesinthesamepartarealsointhesamepartoftherenement. Adashdottededge, ,willrepresentanorangeedgeinthe4edgecoloredcomplete graph. Theconstraintsforthis4edgecolormodelusemanyoftheformerconstraints. Figure2.5presentsconstraintsthatforcetherenementofa4partitegraphinto a12partitegraph.However,theseconstraintsneithergivetheadjacencystructure 26
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Constraint MeaninginCompleteGraphModel ForcedGraphProperty =0 nored K 3 no K 3 =0 neartransitivityofgreenedges multipartiteinred/blue =0 no K 5 inredandblue 4partiteinred/blue + = 1 4 1 4 ofedgesaregreen/orange balanced4partiteinred/blue =0 transitivityoforangeedges multipartiteinred/blue/green =0 no K 4 ingreen 3partiteingreen Figure2.5:Flagalgebraconstraintsfor4edgecoloredcompletegraphmodelinga 12partite, K 3 freegraph. picturedinFigure2.1norgivethebalancednatureofSperfeld's[84]optimalconstruction.Implementingthis4edgecolormodeldidnotobtainbetterresultsthan the3edgecolormodel.Forthemotivatedreader,wenotethatitmaybepossible toimproveourboundusingthe4edgecolormodelbyaddingconstraintstobalancethegraph;wedidnotexertmucheorttowardestablishingtheseagalgebra constraints. 27
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CHAPTER3 INDUCEDCYCLES Inthischapter,weconsideraninducibilityproblemforcyclesinwhichwewill maximizethenumberofinducedcyclesinagraphoforder n .Foragraph H on k vertices,let H n denotethemaximumnumberofinducedcopiesof H inan n vertex graph.SimilartoProposition1.1,wehavethefollowingpropositionregarding H n : Proposition3.1 Forany k vertexgraph H andany n k H n +1 )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n +1 k H n )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n k : Proof: Let G beagraphon n +1 verticeswith H n +1 inducedcopiesof H Sinceeachcopyof H in G has k verticesand G has n +1 vertices, G musthavea vertex v inatmost k H n +1 n +1 copiesof H .In G )]TJ/F20 11.9552 Tf 11.955 0 Td [(v ,thereareatleast H n +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k H n +1 n +1 = n +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k H n +1 n +1 copiesof H .However,the n vertexgraph G )]TJ/F20 11.9552 Tf 10.495 0 Td [(v hasatmost H n copiesof H .Thus, n +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k H n +1 n +1 H n : Usingthecombinatorialequality n +1 k = n +1 n +1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(k n k ; wehave H n +1 )]TJ/F21 7.9701 Tf 5.48 4.378 Td [(n +1 k H n )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n k ; asdesired. The inducibility of H isdenedtobe i H :=lim n !1 H n )]TJ/F21 7.9701 Tf 5.479 4.378 Td [(n k : 28
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Since H n isnonincreasingbyProposition3.1andisboundedbelowby0,thislimit exists. The balancediteratedblowup ofagraph H isablowupof H whereeveryblowup setbothdiersinsizebyatmostonevertexandalsoinducesabalancediterated blowupof H .Anexampleoftheiteratedblowupof C 5 isprovidedinFigure3.1. Figure3.1:Progressionofobtainingabalanced,iteratedblowupof C 4 In1975,PippengerandGolumbic[74]notedthatthisbalancediteratedblowupforanygraph H on k verticesgivesagenerallowerboundof i H k k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(k Giventhisgenerallowerboundforinducibility,anaturalendeavoristodetermine forwhichgraphs H theinducibilityiswitnessedbythebalancediteratedblowupof H .Equivalently,wemayaskforwhichgraphs H thebalancediteratedblowupof H isanextremalgraphfor i H .Inarandomgraphsetting,Fox,Huang,andLee [44]recentlyannouncedthattheinducibilityofalmostallgraphsiswitnessedbyits balanced,iteratedblowup.However,thisiscertainlynotalwaysthecase.Answering aquestionofBollobs,Egawa,Harris,andJinfrom[16],in[53]Hatami,Hirst,and Norineshowthattheinducibilityofsucientlylargebalancedblowupsisattained byablowupratherthananiteratedblowup.Further,thecollectiveexaminationof theinducibilityofsmallgraphs[16,17,24,41,58]alsoshowsthisisnotthecasefor allgraphsonatmost4verticesexceptcompleteandemptygraphs.Aniceoverview oftheinducibilityofsmallgraphsappearsin[40]. Intheir1975paper,PippengerandGolumbicmadethefollowingconjecture: Conjecture3.2Pippenger,Golumbic[74] For k 5 i C k iswitnessedby 29
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thebalancediteratedblowupof C k Foralongtimeitwasbelievedthatprogressonthisconjecturewasintractableeven for C 5 .WhenRazborov'stheoryofagalgebraswasintroducedover30yearsafter theconjecture,manythoughtthispowerfulnewtoolcouldbeusedtonallymake progressontheconjecture.Whilethisistrue,muchlikethemultipartiteproblem inthelastchapter,thestraightforwardapplicationofCauchySchwarzusingag algebrasunderthestandardapproachdonotgiveupperboundssucientlycloseto theconjecturedlowerboundforstabilityargumentstoestablishexactresults. Inthecaseof C 5 ,theiterativenatureoftheextremalconstructioniswhatcauses problemsforthestandardagalgebraapproach.Whiletherearemanygraphswhose conjecturedextremalconstructionisaniteratedblowup[73],wearenotawareof manyapplicationsofagalgebraswhichcompletelydetermineaniterativestructure. FalgasRavryandVaughan[42]provedaniterativeextremalstructureforsmallorientedstarsthatwasgeneralizedtoallstarsbyHuang[61],thoughheuseddierent methods.Recently,Hladk,Krl,andNorine[60]announcedaresultonoriented pathsoflength2. Inthenextsection,wewilldiscussthepaperofBalogh,Hu,Lidick,andPfender [11]conrmingConjecture3.2for k =5 forgraphsoforder 5 m forsome m 2 N BeyondtheirsubmittedpaperwithVolecandYoung[12]usingasimilarapproach, weareunawareofanyfurtheriterativeextremalstructuresprovedusingagalgebras. TheirgeneralapproachwillbeusedinourresultsinSection3.2. 3.1Induced5Cycles Asmentionedpreviously,Conjecture3.2wasconrmedfor k =5 ongraphswith theappropriatenumberofverticesbyBalogh,Hu,Lidick,andPfender[11].The methodusedisslightlydierentfromthestandardapproach,whichwebrieyreview here.Inthestandardapproach,agalgebracomputationsprovideanupperboundfor i C 5 closetothelowerboundprovidedbythebalancediteratedconstruction.From 30
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thesolutiontothesemideniteprogramestablishedfromagalgebrainequalities, weidentifysubgraphsthatappearwithverylowprobabilityandthenusethisfact withinstabilityargumentstoobtainanextremalconstruction. Thespecicshortcomingofthestandardapproachinapplicationtoshortinducedcyclesarisesintheupperboundobtainedfromagalgebras.Specically,the lowerboundfor i C 5 obtainedfromtheiterativeextremalconstructionandthecorrespondingupperboundobtainedusingagalgebracomputationswithagson7 verticesareasfollows: 0 : 03846153 ::: = 1 26 i C 5 0 : 03846157 Theseboundsleaveagapofapproximately 4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 .Whilethisgapseemsverysmall itisstilltoolargeforthestandardapproachtoovercomegiventheiterativestructure oftheextremalconstruction.Inlargepart,thisiscausedbysmallnumericalerrors havingalargerelativeeectonthedensityofsubgraphsappearingwith0density. Thesesmallnumericalerrorshavelessrelativeeectonthedensityofsubgraphs appearingwithhighdensity.Theiterativemethodcapitalizesonthislesserimpactby usingsuchsubgraphsinstead.Theprevalenceofthesehighdensitysubgraphsprovide informationaboutthegeneraltoplevelstructureoftheextremalconstruction.For induced C 5 s,thehighdensitysubgraphsguaranteeenoughedgesbetweenpartssothat theextremalgraphlookslikethebalancedblowupof C 5 withextraedgespossible ineachpart. Fortheiteratedapproach,acollectionofhighdensitysubgraphsareidentied usingagalgebracomputations.Thehighdensitysubgraphsusedarespecicallythe familyofall7vertexgraphsobtainedfroma C 5 byeitherduplicatingtwovertices onceeach,orduplicatingasinglevertextwice.Stabilityargumentsareusedtoshow thatthistoplevelstructureisinfactabalancedblowupof C 5 andtheninductionis usedtoconrmtheiteratedstructureoftheextremalconstruction.Forourresultin thenextsection,weusethefamilyofall8vertexgraphsobtainedfroma C 6 inthe 31
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similarway. 3.2Induced6Cycles Webeginbydeningsomenotation.Let C k 6 denotethe k )]TJ/F15 11.9552 Tf 12.145 0 Td [(1 timesiterated blowupof C 6 .Inthisnotation, C 1 6 issimply C 6 .Let G n bethesetofallgraphs on n vertices,andlet C 6 G denotethenumberofinducedcopiesof C 6 in G .Dene C 6 n =max G 2G n C 6 G .Agraph G 2G issaidtobe extremal if C 6 G = C 6 n .If n is apowerof m ,wecanexactlydeterminetheuniqueextremalgraphandthus C 6 n Theorem3.3 For k 1 ,theuniqueextremalgraphin G 6 k is C k 6 Figure3.2:Theextremalgraphfor i C 6 C k 6 Toprovetheiterativestructureof C k 6 inTheorem3.3,werstprovethefollowing theoremaboutthetoplevelstructureof C k 6 Theorem3.4 Thereexists n 0 suchthatforevery n n 0 C 6 n = 5 Y i =0 x i + 5 X i =0 C 6 x i ; where P 5 i =0 x i = n and x 0 ;x 1 ;:::;x 5 areasequalaspossible. Moreover,if G 2G n isanextremalgraph,then V G canbepartitionedintosets X 0 ;X 1 ;:::;X 5 ofsizes x 0 ;x 1 ;:::;x 5 respectively,suchthatfor 0 i
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3.2.1TopLevelStructureoftheExtremalGraph Weconsiderdensitiesof8vertexsubgraphs.Basedontheconjecturedextremal graph,weconsidertwofamiliesofgraphsthatareratherprevalent.Inparticular,let C 22 bethefamilyofgraphsthatcanbeobtainedfrom C 6 byduplicatingtwovertices and C 3 bethefamilyofgraphsthatcanbeobtainedfrom C 6 byduplicatingasingle vertextwice.Forbothfamilies,thepresenceorabsenceofedgesbetweentheoriginal verticesandtheirduplicatesspecifygraphsinthefamily,seeFigure3.3. Figure3.3:Representativegraphsfor C 22 and C 3 wheredottededgesdenotethe possibilityofedges.Thefamily C 22 consistsof9graphsandthefamily C 3 consists of4graphs. Welet C 22 representanygraphin C 22 and C 3 representanygraphin C 3 .For convenience,wewillalsouse C 22 and C 3 todenotethedensitiesoftheserespective graphs,thatistheprobabilitythatrandomlyselecting8verticesinducesthecorresponding8vertexexpansionof C 6 .Additionally,forasetofvertices Z C 22 Z and C 3 Z denotetherespectivedensityof C 22 and C 3 containing Z .Thatis,fora graph G on n vertices C 22 Z and C 3 Z aretherespectivenumberof C 22 and C 3 containing Z dividedby )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n j Z j 8 j Z j Notethatfor C k 6 inthelimitas k !1 wehave 4 C 22 )]TJ/F15 11.9552 Tf 12.579 0 Td [(15 C 3=0 .Similar to[11],weneedapositivelowerboundforourapproachtowork.Inwhatfollows, a =4 : 95 .Thelimitvaluesofinterestinthebalancediteratedblowup C k 6 ,where 33
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k !1 ,are C 6 = 24 1555 0 : 0154340836 ; and C 22 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 a 4 C 3= 5040 55987 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(3 a 4 1344 55987 0 : 00090020897 : Weobtaincomparablevaluesusingagalgebracomputations. Proposition3.5 Thereexists n 0 suchthateveryextremalgraph G onatleast n 0 verticessatises C 6 5403179531366257035327 350000000000000000000000 < 0 : 015437656 ,and C 22 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(3 a 4 C 3 35999533394428927782939 40000000000000000000000000 > 0 : 0008999883348 : Proof: Weapplystandardagalgebracomputations.Therstinequalityisobtaineddirectlywithcomputationsonsevenvertices.Thesecondinequalityisobtained byminimizing C 22 oneightverticeswiththeaddedconstraintthat C 6 24 1555 Itisworthmentioningthatagalgebraboundsfromcomputingon8vertices obtainbetterboundson C 6 .However,theseboundsarenotsucientforthestandard agalgebraapproachtobesuccessful. Let G beanextremalgraphon n verticeswith n n 0 fromProposition3.5.Let Z bethefamilyofallinduced C 6 s.Let Z = z 0 z 1 z 5 beaninduced C 6 maximizing C 22 Z )]TJ/F20 11.9552 Tf 11.956 0 Td [(a C 3 Z .Then, )]TJ/F32 11.9552 Tf 5.479 9.683 Td [(C 22 Z )]TJ/F20 11.9552 Tf 11.955 0 Td [(a C 3 Z n )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 2 1 jZj X Y 2Z )]TJ/F32 11.9552 Tf 5.479 9.683 Td [(C 22 Y )]TJ/F20 11.9552 Tf 11.955 0 Td [(a C 3 Y n )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 2 )]TJ/F15 11.9552 Tf 5.48 9.684 Td [(4 C 22 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 a C 3 )]TJ/F21 7.9701 Tf 10.959 4.379 Td [(n 8 C 6 )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n 6 = 4 C 22 )]TJ/F15 11.9552 Tf 11.956 0 Td [(3 a C 3 28 C 6 n )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 2 : WithProposition3.5and a =4 : 95 ,wehave C 22 Z )]TJ/F20 11.9552 Tf 11.956 0 Td [(a C 3 Z 4 0 : 0008999883348 28 0 : 015437656 > 0 : 008328321483 : .1 34
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Fortheremainderofthissection,unlessotherwisespecied,indicesandarithmeticthereonarein Z 6 .Wedenesetsofvertices Z i thathavethesameneighbors in Z as z i .Formally, Z i := f v 2 V G : G [ Z n z i [ v ] = C 6 g : Noticethat Z i Z j = ; for i 6 = j .Wecallapair v i v j funky if v i v j isanedgewhile z i z j isnotanedgeorviceversa,with i 6 = j v i 2 Z i and v j 2 Z j .Noticethat G [ Z [f v i ;v j g ] C 22 ,whichmeanseveryfunkypairdestroysapotentialcopyof C 22 Z .Let E f denotethesetoffunkypairs.Thus,.1implies X i 0 : 008328321483 n )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 2 : Foranychoiceofdisjointsets X i V G ,let T := V G n S X i .Toreectthe moregeneralchoiceofsetsweadjustourdenitionofafunkypairtoapair v i ;v j with v i 2 X i ;v j 2 X j where v i v j isanedgewhen i )]TJ/F20 11.9552 Tf 12.25 0 Td [(j 62f 1 ; 5 g orwhen v i v j isnot anedgewhen i )]TJ/F20 11.9552 Tf 12.434 0 Td [(j 2f 1 ; 5 g .Inthefollowingdenitionweslightlyabusenotation byusing E f toreectouradjusteddenitionoffunkypairs.Fornormalization,let f = j E f j n 2 x i = 1 n j X i j ,and t = 1 n j T j .Choosesets X i Z i possibly X i = Z i suchthat thelefthandsideof 2 X i 0 : 008328321483 .2 ismaximized. Claim3.6 Thefollowinginequalitiesaresatised: 0 : 16582
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t + P i x i =1 ,and x i ;f;t 0 ineachcase.Bysymmetrytheboundsfor x 0 holdfor all x i Herewedescribeobtainingthelowerboundon x 0 in.3.Weneedtosolvethe followingprogram P : P 8 > > > > > > > > > > < > > > > > > > > > > : minimize x 0 subjectto t + P i x i =1 ; 2 P i > > > > > > > > > < > > > > > > > > > > : minimize x 0 subjectto t + x 0 +5 y =1 ; 10 x 0 y )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 95 5 y 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 : 95 x 2 0 0 : 008328321483 ; x 0 ;y;t 0 : 36
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Wesolve P 0 usingLagrangemultipliers.ThisworkisdelegatedtoSage[30]andthe Sagescriptisprovidedasasupplementallenamed solvexi.sage .Sincending anupperboundfor x 0 occursonthesameconstraints,thesamescriptcanbeused simplybychangingtheobjectivetomaximization. Weturnourattentiontondingtheupperboundfor f in.4.Since f isonly constrainedby.2,wecanmaximize f bymaximizing 2 X i > > > > > > > > > < > > > > > > > > > > : maximize f subjectto 1 t 0 ; 0 : 3 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(t 6 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 f 0 : 008328321483 ; f 0 : Wesolve P 00 usingLagrangemultipliersbymakingtheappropriatechangestothe Sagecode. Weproceedsimilarlyforanupperboundon t .Since t isonlyconstrainedby t + P x i =1 ,thelefthandsideof.2ismaximizedwhen f =0 and x i = x j ,as before.Substituting x i = 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(t 6 gives P 000 8 > > > > > > < > > > > > > : maximize t subjectto 1 t 0 ; 0 : 3 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(t 6 2 0 : 008328321483 : MakingtheappropriatechangestotheSagecodeenablessolving P 000 usingLagrange multipliers. 37
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Foranyvertex v 2 X i ,weuse d f v todenotethenumberoffunkypairsincident to v afternormalizingby n ,thatisthe funkydegree of v .Ifwemove v 2 X i to T thenthelefthandsideof.2willdecreaseby 1 n 2 X j 6 = i x j )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d f v )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 4 : 95 x i + o : Ifthisquantitywasnegative,thenthelefthandsideof.2couldbeincreasedby moving v to T ,whichwouldcontradictourchoiceof X i .Togetherwith.3,this implies d f v X j 6 = i x j )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 : 95 x i + o 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 : 95 x i + o 0 : 013371 ; .6 whichwewillusetoobtainacontradictioninClaim3.9.Foruseinobtainingthis contradiction,werstincludeabriefargumentregardingthenumberoffunkyedges inaninduced C 6 Claim3.7 Anyinduced C 6 in S X i usingafunkypairmustuseatleasttwofunky pairs. Proof: Supposeforcontradictionthat C = v 0 v 1 v 5 isaninduced C 6 using exactlyonefunkyedge.Withoutlossofgenerality,let v 0 beincidenttothisfunky edge.Noticethat C )]TJ/F20 11.9552 Tf 11.653 0 Td [(v 0 isaninduced P 5 .Sincetherearenofunkyedgesin C )]TJ/F20 11.9552 Tf 11.654 0 Td [(v 0 C )]TJ/F20 11.9552 Tf 11.463 0 Td [(v 0 musthaveeitheratmostonevertexineach X i oreveryvertexinasingle X i Assume C )]TJ/F20 11.9552 Tf 10.665 0 Td [(v 0 hasatmostonevertexineachpart.Since C usesafunkyedge, v 0 mustbeinthesamepartasanother v i for i 2f 1 ;:::; 5 g .If v 0 isinthesamepartas some v i for i 2f 1 ; 2 ; 3 g ,then v 0 v 5 and v 0 v i +1 arebothfunkyedges,acontradiction. Similarly,if v 0 isinthesamepartassome v i for i 2f 4 ; 5 g ,then v 0 v 1 and v i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 are bothfunkyedges,acontradiction. Thus,everyvertexof C )]TJ/F20 11.9552 Tf 12.238 0 Td [(v 0 isinthesamepart,say X i .Since C usesafunky edge, v 0 cannotbein X i .If v 0 2 X i )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 [ X i +1 ,then v 0 v 2 ;v 0 v 3 ;v 0 v 4 arefunkyedges 38
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usedby C ,acontradiction.If v 0 62 X i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 [ X i +1 ,then v 0 v 1 ;v 0 v 5 arefunkyedgesused by C ,acontradiction.Therefore, C cannotuseexactlyonefunkyedge. Forbrevity,welet d f denote max f d f v : v 2[ X i g .Similarly,welet x max and x min denote max f x i : i 2 Z 6 g and min f x i : i 2 Z 6 g ,respectively.Forclarityin showingtherearenofunkypairs,werstpresentaparticularlydetailedcase.Should thetrustingreaderwishtoskipthedetails,theproofofTheorem3.4continueson page44. Claim3.8 Thereareatmost tn 10 x max n 3 induced C 6 scontainingbothexactlyone vertexfrom T andafunkypair uv Proof: Consideringthata C 6 withoutitsvertexfrom T isaninduced P 5 using exactlyonefunkypair,weexaminethewaystoembedsucha P 5 .Wearguethat therearenomorethan t 10 x 3 max n 4 such C 6 sinthreecasesbasedonthetypeoffunky pairused. Thefollowinglimitationonchoosingadditionalverticesinbuildinga P 5 applies toallthreecases,soweincludeithere.If X i hasavertexandwepick2additional verticesfrom X i +1 touseintheinduced P 5 ,thenwehaveaninduced P 3 whose endpointshavethesameadjacencypatterntoverticesoutsideof X i +1 .However, pickingathirdadditionalvertexfrom X i +1 forcesthevertexin X i tohave3neighbors andpickingathirdadditionalvertexfrom X i [ X i +2 createsa C 4 .Sinceasimilar argumentholdsifwepicked2additionalverticesfrom X i )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ,wecanpickatmost1 additionalvertexineach X i Case1: u 2 X i v 2 X j with i )]TJ/F20 11.9552 Tf 11.955 0 Td [(j 2f 1 ; 5 g Withoutlossofgenerality,let u 2 X 0 and v 2 X 1 .Webeginbylimitingwhere andhowmanyverticescanbechosenfrom X i s. Becauseaninduced P 5 with uv asitsonlyfunkypaircannothave u and v as endpoints,wemustuseatleastoneadditionalvertexin X 0 [ X 1 .However,using 39
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anadditionalvertexfromboth X 0 and X 1 createsaninduced P 4 oragraphwith acycle.Assumingthischoiceofadditionalverticesgivesaninduced P 4 ,usingan additionalvertexfrom X 5 or X 1 createsavertexofdegree3in X 0 andusingan additionalvertexfrom X 0 or X 2 createsavertexofdegree3in X 1 .Becausewecan notextendthis P 4 toaninduced P 5 ,wecanuseatmostoneadditionalvertexfrom X 0 [ X 1 Withtheabovelimitations,therearefourwaystopick3additionalverticesto extend uv toaninduced P 5 byrstpickinganadditionalvertexfrom X 0 or X 1 and thenconsideringwhetherornotitinducesanedgewithinthat X i .Thesefourinduced P 5 sarepicturedinFigure3.4.Thus,thereareatmost tn 4 x max n 4 C 6 sinthiscase. X 0 X 1 X 2 X 3 X 4 X 5 u v X 0 X 1 X 2 X 3 X 4 X 5 u v X 0 X 1 X 2 X 3 X 4 X 5 u v X 0 X 1 X 2 X 3 X 4 X 5 u v Figure3.4:Theembeddingsofaninduced P 5 withonefunkyedgein G [ X 0 ;X 1 ] Case2: u 2 X i v 2 X j with i )]TJ/F20 11.9552 Tf 11.955 0 Td [(j 2f 2 ; 4 g Forthiscase,noticethatwecannotpickanadditionalvertexfrom X 1 without creatinga C 3 .Weexaminefoursubcasesbasedonthenumberofadditionalvertices usedin X 0 .EachofthesubsequentpossibilitiesarepicturedinFigure3.5. Ifthreeadditionalverticesarepickedin X 0 ,thenwehaveonewayaninduced P 5 canappear.Iftwoadditionalverticesarepickedin X 0 ,thethirdadditionalvertex mustappearineither X 2 or X 3 Assumethatonly1additionalvertexispickedin X 0 .Ifwepick1additional vertexin X 5 themostwecouldpossiblypick,thentheremainingadditionalvertex mustbepickedineither X 2 or X 3 .Ifwedonotpickanadditionalvertexin X 5 ,then thetworemainingadditionalverticesmustbepickedrelativeto X 2 ,inwhich0,1,or 40
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X 0 X 1 X 2 X 3 X 4 X 5 u v a X 0 X 1 X 2 X 3 X 4 X 5 u v b X 0 X 1 X 2 X 3 X 4 X 5 u v c X 0 X 1 X 2 X 3 X 4 X 5 u v d X 0 X 1 X 2 X 3 X 4 X 5 u v e X 0 X 1 X 2 X 3 X 4 X 5 u v f X 0 X 1 X 2 X 3 X 4 X 5 u v g X 0 X 1 X 2 X 3 X 4 X 5 u v h X 0 X 1 X 2 X 3 X 4 X 5 u v i X 0 X 1 X 2 X 3 X 4 X 5 u v j X 0 X 1 X 2 X 3 X 4 X 5 u v k X 0 X 1 X 2 X 3 X 4 X 5 u v l Figure3.5:Theembeddingsofaninduced P 5 withfunkyedgein G [ X 0 ;X 1 ] 2additionalverticesuniquelydetermineaninduced P 5 Finally,assumethatnoadditionalverticesarepickedin X 0 .If X 5 has1additional vertex,thenpickinganadditionalvertexin X 4 forcestheremainingadditionalvertex tobein X 2 ,otherwisewehavea C 5 .If X 4 doesnothavethesecondadditionalvertex, thenwemusthaveatleastoneadditionalvertexin X 2 ,otherwisewecreatea C 5 Thus,thereareeitheroneortwoadditionalverticesin X 2 ,bothofwhichuniquely determinea P 5 Thenalsubcaseisif X 5 has0additionalvertices.Again,thisuniquelydeterminesa P 5 becausewemustpick2additionalverticesfromthesamesettoavoida C 5 and X 2 istheonlysuchsetwherethisispossible. Aninitiallookatcountingthese C 6 sgivesanupperboundof tn 12 x max n 3 41
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However,wecanimprovethisupperboundbyaccountingforthenecessaryadjacencieswhenpickingatleast2additionalverticesfromthesamepart.Takingthisinto account,wehavethatFigures3.5aand3.5lcanbeinatmost tn )]TJ/F21 7.9701 Tf 6.675 4.976 Td [(x max n 3 3 C 6 sand thatFigures3.5b,3.5c,3.5f,and3.5kcanbeinatmost tn )]TJ/F21 7.9701 Tf 6.675 4.976 Td [(x max n 2 2 x max nC 6 s. Thus,wehaveatmost tn 2 1 27 +4 1 4 +6 x max n 3
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X 0 X 1 X 2 X 3 X 4 X 5 u v a X 0 X 1 X 2 X 3 X 4 X 5 u v b X 0 X 1 X 2 X 3 X 4 X 5 u v c X 0 X 1 X 2 X 3 X 4 X 5 u v d X 0 X 1 X 2 X 3 X 4 X 5 u v e X 0 X 1 X 2 X 3 X 4 X 5 u v f X 0 X 1 X 2 X 3 X 4 X 5 u v g X 0 X 1 X 2 X 3 X 4 X 5 u v h X 0 X 1 X 2 X 3 X 4 X 5 u v i X 0 X 1 X 2 X 3 X 4 X 5 u v j X 0 X 1 X 2 X 3 X 4 X 5 u v k X 0 X 1 X 2 X 3 X 4 X 5 u v l X 0 X 1 X 2 X 3 X 4 X 5 u v m X 0 X 1 X 2 X 3 X 4 X 5 u v n X 0 X 1 X 2 X 3 X 4 X 5 u v o X 0 X 1 X 2 X 3 X 4 X 5 u v p Figure3.6:Theembeddingsofaninduced P 5 withfunkyedgein G [ X 0 ;X 3 ] 43
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Iftherearenoadditionalverticespickedfrom X 0 ,thenall3additionalvertices mustbepickedfrom X 3 ,otherwisethereisa C 3 or C 4 .Thisuniquelydeterminesa P 5 AsbeforeinCase2,wecanimproveupontheinitialcountof tn 16 x max n 3 by accountingforwhenmorethanoneadditionalvertexispickedfroman X i .Since Figures3.6iand3.6phave3additionalverticesinthesamepartandFigures3.6d, 3.6h,3.6j,3.6k,3.6l,and3.6ohave2additionalverticesinthesamepart,wehaveat most tn 2 1 27 +6 1 4 +8 x max n 3
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induced C 6 sbypickingthetworemainingverticesfromdistinct X i samongthetwo X i s notcontaining u;v;u 0 ;v 0 .Thus G 0 hasatleast x 4 min )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f max x 2 max )]TJ/F20 11.9552 Tf 11.955 0 Td [(fx 3 max n 4 .7 induced C 6 scontaining uv Forcomparison,wecountthenumberofinduced C 6 sin G containing uv .Claim 3.7guaranteesanysuch C 6 includesasecondfunkypair.Thenumberof C 6 scontaininganotherfunkypair u 0 v 0 with f u;v gf u 0 ;v 0 g = ; canbegenerouslyupperbounded bypickinganyfunkypairandtwoadditionalvertices,thatis fn 2 n 2 .Thus,inthe subsequentboundswecanassumethatallfunkypairsareincidentto uv Ifthereareatleasttwosuchfunkypairsinaninduced C 6 ,thenwecangenerously upperboundthenumberof C 6 sbyassumingeverysuchpairofincidentfunkyedges createsa C 6 .Thatis,thereareatmost )]TJ/F18 7.9701 Tf 5.479 4.379 Td [( d f v + d f u n 2 n 2 C 6 swithatleasttwofunky pairsincidentto uv Ifthereisonlyonefunkypairincidentto uv inaninduced C 6 ,weclaimthat eitherthe C 6 includesavertexfrom T orthechoiceoftheremaining3verticesinthe C 6 areforced.Intherstcase,wecangenerouslyupperboundthenumberofsuch C 6 sbypickinganincidentfunkyedge,avertexfrom T ,andthenanytwoadditional vertices.Thisgivesanupperboundof )]TJ/F20 11.9552 Tf 5.479 9.684 Td [(d f v + d f u n tn n 2 Let C beaninduced C 6 containing uv thatdoesnotincludeavertexfrom T and hasonlyonefunkypairincidentto uv .Sincewepreviouslycountedallofthe C 6 s withafunkyedgenotincidentto uv ,wecanassumethatallbothfunkypairsin C areincidentto u ,say uv and uv 0 arethetwofunkypairsof C .Withoutlossof generality,let u 2 X i and v 2 X j with i 6 = j .Notethat C )]TJ/F20 11.9552 Tf 12.555 0 Td [(u isaninduced P 5 Thus,if v 0 2 X j ,then V C )]TJ/F20 11.9552 Tf 12.418 0 Td [(u V j .Consequently,thenumberofchoicesfor C isupperboundedby d f u n x j n 3 ,whichwewilllaterdoubletoaccountforthe choicebetween u and v Ontheotherhand,assumethat v 0 2 X k with 0 6 = k 6 = j .Wewillshowthatthe 45
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choicesfortheremainingthreeverticesof C areforcedintodistinct X i s,whichgives anupperboundof d f u n x max n 3 .Theargumentforthisupperboundispresented inthefollowingthreeparagraphsshouldthetrustingreaderwanttoreferenceFigure 3.7andskipthecaseanalysis. X 0 X 1 X 2 X 3 X 4 X 5 u X 0 X 1 X 2 X 3 X 4 X 5 u v v 0 X 0 X 1 X 2 X 3 X 4 X 5 u v v 0 X 0 X 1 X 2 X 3 X 4 X 5 u v v 0 X 0 X 1 X 2 X 3 X 4 X 5 u v v 0 X 0 X 1 X 2 X 3 X 4 X 5 u v v 0 Figure3.7:Induced C 6 swithallfunkyedgesincidenttoasinglevertex. First,assume u;v 2 X 1 .Since C )]TJ/F20 11.9552 Tf 12.72 0 Td [(u isaninduced P 5 with u incidenttoits endpointsand v 0 notinthesame X j as v ,then v mustbeaninternalvertexofthis path.Becausetwoconsecutiveverticesofaninduced P 5 inthesame X i wouldcreate a C 3 ,eachvertexofthis P 5 mustbeindistinct X i s.Considering v isaninternal vertex,therearethreepotentialoptionsforthechoiceof P 5 : v 0 vv 2 v 3 v 4 v 5 v 0 vv 2 v 3 and v 4 v 5 v 0 vv 2 ,where v ` 2 X ` .Noticethatthethirdpossibilitywouldrequirean additionalfunkypairforaninduced P 5 .Hence,wedonotincludeitinthiscount. Thechoiceofthefunkypairincidentto u givingriseto C gives v 0 ineither X 3 or X 4 ,whichdeterminesthechoicesoftheremainingthreeverticesinto X 5 ;X 0 ;X 2 or X 0 ;X 2 ;X 3 ,respectively.Asimilarargumentfor v 2 X 5 holdsbyreplacingindices withtheiradditiveinversein Z 6 46
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Next,assume u 2 X 0 and v 2 X 2 .Since uv isanedge, v mustbeanendpoint oftheinduced P 5 C )]TJ/F20 11.9552 Tf 12.453 0 Td [(u .Asbeforeinregardtoconsecutivevertices, C )]TJ/F20 11.9552 Tf 12.453 0 Td [(u spans either X 2 X 3 X 4 X 5 X 0 with v 0 2 X 5 ,or X 4 X 5 X 0 X 1 X 2 .Notethatthesecondoptionis notpossibleasitrequiresthreefunkypairsbeincidentto u .Thus,thechoiceofa funkypairincidentto u givingriseto C gives v 0 2 X 5 ,whichdeterminesthechoices oftheremainingthreevertices.Asbefore,asimilarargumentfor v 2 X 4 holdsby replacingindiceswiththeiradditiveinversein Z 6 Finally,assume u 2 X 0 and v 2 X 3 .As uv isanedge, v mustbeanendpointof theinduced P 5 C )]TJ/F20 11.9552 Tf 10.844 0 Td [(u .Asinthetwopreviouscases, C )]TJ/F20 11.9552 Tf 10.843 0 Td [(u spanseither X 3 X 4 X 5 X 0 X 1 with v 0 2 X 5 or X 5 X 0 X 1 X 2 X 3 with v 0 2 X 1 .Thus,thechoiceofafunkypairincident to u givingriseto C gives v 0 2 X 1 [ X 5 andthechoicesfortheremainingthreevertices areforcedintotheappropriate X i s. Wenowconsidertheremainingtwocaseswhentherearenofunkypairsincident to uv ,bothofwhichuseverticesfrom T .Ifthereareatleasttwoverticesfrom T thenwecangenerouslyupperboundthenumberofinduced C 6 s containing uv by pickingtwoverticesfrom T andanytwoadditionalvertices.Thisgivesatmost )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(tn 2 n 2 such C 6 s.Ifthereispreciselyonevertexfrom T inaninduced C 6 containing uv ,then Claim3.8gives tn 10 x max n 3 asanupperbound. Combiningandcomparingallofthecountingboundswithinthisclaim,thenumberofinduced C 6 scontaining uv dividedby n 4 is inG: 2 d f 2 + f +2 d f x 3 max + t +10 tx 3 max + t 2 0 : 000505 inG': x 4 min )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d f x 3 max )]TJ/F20 11.9552 Tf 11.955 0 Td [(d f x 2 max 0 : 000630 : Thiscontradictstheextremalityof G Wehavejustshownthattherearenofunkypairsamong S X i .Infurthercleaning thetoplevelstructureof G ,wewanttoshowthat T = ; .Oncecompletedthiswill showthattheverticesof G canbepartitionedinto6setsthatresembletheblowup 47
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ofa C 6 Supposethereexists x 2 T .Wewillmove x tooneofthe X i suchthat d f x isminimal.Bysymmetrywemayassumethat x isaddedto X 0 .Notethatadding asinglevertexto X 0 doesnotchangeanyofthedensityboundsweusedaboveby morethan o Claim3.10 Forevery x 2 T ,if x isaddedto X 0 ,then d f x 0 : 0481706 Proof: Let xw beafunkypair.Let G 0 beobtainedfrom G byxing xw through addingordeletingtheedge xw .Since G isextremal,wehave C G 0 C G .The analysisforthisproofissimilartothatinClaim3.9.However,wecansayabitmore sinceeveryfunkypaircontains x First,wecounttheinduced C 6 scontaining xw in G .Wegenerouslygivean upperboundforthenumberof C 6 s containing xw andanothervertexfrom T by selectingavertexfrom T andanythreeremainingvertices,thatis tn n 3 .Asin Claim3.9,anyinduced6cycle C containing x has C )]TJ/F20 11.9552 Tf 12.282 0 Td [(x asaninduced P 5 .Citing thepreviousanalysisandFigure3.7,thenumberofinduced C 5 scontaining xw isat most )]TJ/F20 11.9552 Tf 5.479 9.684 Td [(d f x n x max n 3 In G 0 thereareatleast x min n 4 )]TJ/F26 11.9552 Tf 11.72 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 9.684 Td [(d f x n x min n 3 induced C 6 scontaining xw Thus,wehave C G n 4 d f x x 3 max + t; and C G 0 n 4 x 4 min )]TJ/F20 11.9552 Tf 11.955 0 Td [(d f x x 3 min : Since C G 0 C G ,wehave x 4 min )]TJ/F20 11.9552 Tf 11.955 0 Td [(d f x x 3 min d f x x 3 max + t; whichtogetherwith.3and.5gives d f x 0 : 0481706 Forvertices u;v 2 V G ,let C u 6 denotethenumberof C 6 sin G containing u and C uv 6 denotethenumberof C 6 scontainingboth u and v 48
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Claim3.11 Everyvertexoftheextremalgraph G isinatleast )]TJ/F18 7.9701 Tf 10.909 4.977 Td [(24 1555 + o )]TJ/F21 7.9701 Tf 12.952 4.379 Td [(n 5 0 : 000128617 n 5 induced C 6 s. Proof: Notethatatrivialboundis C uv 6 )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 4 .Consider G 0 obtainedfrom G by deleting v andaddingacopyof u .Since G isextremal,wehave 0 C G 0 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C G C u 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C v 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C uv 6 C u 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C v 6 )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( n )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 4 : Thus, C u 6 )]TJ/F20 11.9552 Tf 11.956 0 Td [(C v 6 n 4 Supposeforcontradictionthatavertexof G isinlessthan )]TJ/F18 7.9701 Tf 10.909 4.977 Td [(24 1555 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o )]TJ/F21 7.9701 Tf 12.951 4.379 Td [(n 5 induced C 6 s.Since C u 6 )]TJ/F20 11.9552 Tf 10.994 0 Td [(C v 6 n 4 forall u;v 2 V G ,everyvertexof G isinlessthan )]TJ/F18 7.9701 Tf 10.909 4.977 Td [(24 1555 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o )]TJ/F21 7.9701 Tf 12.952 4.379 Td [(n 5 + n 4 induced C 6 s.Summingoverallverticesof G anddividingby6 toadjustforovercount,wehavethat G containslessthan n 6 24 1555 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o n 5 + n 4 24 1555 )]TJ/F20 11.9552 Tf 11.955 0 Td [(o n 6 induced C 6 s.Since C k 6 has )]TJ/F18 7.9701 Tf 10.909 4.976 Td [(24 1555 + o )]TJ/F21 7.9701 Tf 12.951 4.378 Td [(n 6 induced C 6 s, G isnotextremal,a contradiction. Havingestablishedthenumberofinduced C 6 severyvertexof G isin,weaimto showthatanyvertexof T isintoofew.Oncethisisshown, G willhavenofunky pairsand T = ; .Notethatthiswillroughlydeterminethetoplevelstructureof G Claim3.12 T isempty. Proof: Assume x 2 T .Wecountthenumberofinduced C 6 scontaining x .Let a i n bethenumberofneighborsof x in X i and b i n bethenumberofnonneighborsof x in X i Thenumberof C 6 scontaining x andveverticesfrom S X i isupperboundedby X i a i b i +1 b i +2 b i +3 a i +4 + 1 12 X i a 2 i b 3 i n 5 ; wherewecountinduced P 5 swhenverticesareindistinctpartsandwhenverticesare allinthesamepart.Thevariables a i ;b i satisfytheupperboundin.3andClaim 3.9. 49
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Moreover,wealsoneedtoincludethecasesthatthe C 6 scancontainverticesfrom T .Tothisend,let an bethenumberofneighborsof x in T and bn bethenumberof nonneighborsof x in T .Wegenerouslyupperboundthenumberof C 6 scontaining verticesof T byincreasingeachofthe a i sand b i sby a and b ,respectively.Thus,we wanttosolvetheprogram P 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : maximize P i a i + a b i +1 + b b i +2 + b b i +3 + b a i +4 + a + 1 12 P i a 2 i b 3 i ; subjectto a + b + P i a i + b i =1 ; 0 : 16583 a i + b i 0 : 16751 ; a + b 0 : 00031 ; b i + a i +1 + a i +2 + a i +3 + b i +4 0 : 048 ; a i ;b i ;a;b 0 ; inwhichthepenultimateconstraintusesClaim3.10inlowerboundingthefunky degreeof x .Insteadofsolving P ,wesolveaslightrelaxation P 0 withincreased upperboundson a i + b i ,whichallowsustodrop a and b .Sincetheobjectivefunction ismaximizing,wecanclaimthat a i + b i isalwaysaslargeaspossible. P 0 8 > > > > > > > > > > < > > > > > > > > > > : maximize h = P a i b i +1 b i +2 b i +3 a i +4 + 1 12 P a 2 i b 3 i ; subjectto a i + b i =0 : 16751+0 : 00031 ; b i + a i +1 + a i +2 + a i +3 + b i +4 0 : 048 ; a i ;b i 0 : Notethattheresultingprogram P 0 hasonly6degreesoffreedom. Wendanupperboundforthesolutionof P 0 throughanaiveoptimization. Werstdiscretizethesetoffeasiblesolutionswithauniformmeshandthenbound thegradientoftheobjectivefunction h toaccountforthebehaviorbetweenmesh points.Forthis,wexaconstant s thatwillcorrespondtothenumberofstepsin eachvariablefrom0to0.1679whengeneratingouruniformmesh,whichincludethe 50
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endpoints.Throughitsconstruction,thismeshincludesallfeasiblesolutionsof P 0 andhenceof P ,inoneofthe6dimensionalmeshboxes. Aftergeneratingourmesh,wemustndthepartialderivativesof h .Since h is symmetric,weonlycheckthepartialderivativewithrespectto a 0 : @h @a 0 = b 1 b 2 b 3 a 4 + a 2 b 3 b 4 b 5 + 1 6 a 0 b 3 0 : Wewanttondanupperboundon @h @a 0 .Picking0.1679asanupperboundfor a i + b i allowsustoassume a 0 + b 0 = a 2 + b 2 = a 3 + b 3 = a 4 + b 4 = b 1 = b 5 =0 : 1679 ; whilewemaximize b 1 b 2 b 3 a 4 + a 2 b 3 b 4 b 5 =0 : 1679 : 1679 )]TJ/F20 11.9552 Tf 11.956 0 Td [(a 3 : 1679 )]TJ/F20 11.9552 Tf 11.955 0 Td [(a 2 a 4 + a 2 : 1679 )]TJ/F20 11.9552 Tf 11.955 0 Td [(a 4 =0 : 1679 : 1679 )]TJ/F20 11.9552 Tf 11.956 0 Td [(a 3 : 1679 a 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 a 2 a 4 +0 : 1679 a 4 : Thisismaximizedif a 2 =0 ;a 4 =0 : 1679 or a 2 =0 : 1679 ;a 4 =0 andgivesthevalue 0 : 1679 4 .Hence, a 0 b 3 0 =27 a 0 b 0 3 3 27 a 0 + b 0 4 4 4 = 27 0 : 1679 4 256 : Theresultingupperboundis @h @a 0 0 : 1679 4 + 2 5 27 0 : 1679 4 256 0 : 0008283 : Hence,ina6dimensionalmeshboxwithsidelength ` ,thevalueof h cannotbebigger thanthevalueatacornerplus 6 ` 2 0 : 0008283 ,wherethefactor 6 ` 2 comesfromthefact thattheclosestmeshpointisadistanceatmost ` 2 ineachofthe6variables. When s =80 ,wecomputethemaximumoverallmeshpointstobeatmost 0.00009983.Alongwiththepreviousarithmeticcomputationsforbounds,thiscan becheckedusingtheC ++ codeprovidedinasupplementallenamed meshopt.cpp With ` = 0 : 1679 80 ,wehave 6 ` 2 0 : 0008283 0 : 0000052152 : 51
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Therefore,weconcludethat x isinatmost : 00009983+0 : 0000052152 n 5 < 0 : 00011 n 5 induced C 6 s,acontradictionwithClaim3.11. Asabriefrecap,wehaveshownthatthevertexsetofof G canbepartitioned into6setsthatresembletheblowupofa C 6 .Withnofunkypairsbetweenthese sets,wehavenearlydeterminedthatthetoplevelstructureoftheextremalgraph G AllthatremainstoshowtoproveTheorem3.4istoshowthatthesetsarebalanced. Tothisend,noticethatinduced C 6 scanonlyappearwithinthisstructureeither byintersectingeach X i exactlyonceorbybeingentirelycontainedwithinasingle X i .Thisimpliesthat C n = n 6 Y i x i + X i C x i n ; where C n denotesthemaximumnumberofinduced C 6 sinagraphoforder n .A directtranslationofProposition3.1gives C n +1 )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(n +1 6 C n )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n 6 .8 forall n .Since C n 0 ` :=lim n !1 C n )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(n 6 exists.Thus, ` + o =6! Y i x i + ` X i x 6 i ; which,withtheconstraintson x i ,gives x i = 1 6 + o and ` = 24 1555 .Itremainsto showthat j X i j)222(j X j j 1 Claim3.13 Forlarge n ,wehave j X i j)222(j X j j 1 forall i;j 2 Z 6 Proof: Supposeforcontradictionthat j X 1 j)237(j X 2 j 2 .Pick u 2 X 1 sothat C u 6 is minimizedoververticesin X 1 andpick v 2 X 2 sothat C v 6 ismaximizedoververtices 52
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in X 2 .Since G isextremal, C u 6 + C uv 6 )]TJ/F20 11.9552 Tf 12.034 0 Td [(C v 6 0 otherwisethenumberof C 6 scanbe increasedbyreplacing u byacopyof v Dene y i := j X i j = x i n .By.8,wehave 24 1555 + o C y 2 )]TJ/F21 7.9701 Tf 5.48 4.379 Td [(y 2 6 C y 1 )]TJ/F21 7.9701 Tf 5.479 4.379 Td [(y 1 6 24 1555 : Usingthat y 1 )]TJ/F20 11.9552 Tf 12.615 0 Td [(y 2 2 andthatremovingavertexfromanyinduced C 6 givesan induced P 5 eitherintersectingexactly1part5timesorintersecting5partsexactly once, C u 6 + C uv 6 )]TJ/F20 11.9552 Tf 11.956 0 Td [(C w 6 C y 1 y 1 + y 2 y 3 y 4 y 5 y 6 + y 3 y 4 y 5 y 6 )]TJ/F20 11.9552 Tf 13.15 8.087 Td [(C y 2 y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 y 3 y 4 y 5 y 6 y 2 C y 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 C y 2 y 1 y 2 + y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 +1 y 3 y 4 y 5 y 6 24 1555 + o 1 y 1 y 2 y 2 y 1 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 y 2 6 + y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 +1 y 3 y 4 y 5 y 6 24 1555 6! + o y 5 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 5 2 + y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 +1 y 3 y 4 y 5 y 6 = 24 1555 6! + o y 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 2 y 4 1 + y 3 1 y 2 + y 1 y 3 2 + y 4 2 + y 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 1 +1 y 3 y 4 y 5 y 6 = y 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y 2 24 1555 6! + o 4 n 4 1296 )]TJ/F20 11.9552 Tf 18.997 8.088 Td [(n 4 1296 + + o n 4 1296 2 24 1555 6! + o 4 n 4 1296 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(+ o n 4 1296 < 0 ; acontradictionwith C u 6 + C uv 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C v 6 0 Thisclaimforcesthesix X i s,whichresembletheblowupofa C 6 ,tobebalanced. Hence,theproofofTheorem3.4iscomplete. 3.2.2IterativeStructureoftheExtremalGraph Havingcompletedthestabilityportionoftheiteratedapproachthatdetermines thetoplevelstructureoftheextremalgraph,weturnourattentiontoinductively provingthattheextremalgraphhasthedesirediteratedstructure.Assuch,Theorem 3.3,whichstatesthat C k 6 istheuniqueextremalconstructionforagraphon 6 k 53
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vertices,isaconsequenceofTheorem3.4.FortheproofofTheorem3.3wewill takeaminimalcounterexampleandshowthatablowupofthisgraphcontradicts Theorem3.4. Aspreviouslymentioned, C 1 6 issimply C 6 .Thus,Theorem3.3isclearlytruefor k =1 Supposeforcontradictionthatthereisagraph G on n =6 k verticeswith C G C C k 6 thatisnotisomorphicto C k 6 .Fix k 2 tobeminimalinthateveryextremal graphon 6 k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 verticesisisomorphicto C k )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 6 If G conformstothestructuredescribedinTheorem3.4,theneachextremal parton 6 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 verticesisisomorphicto C k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 6 bytheminimalityof k .Thismakes G isomorphicto C k 6 .Therefore, V G cannotbepartitionedintosets X i for i 2 Z 6 with j X i j =6 k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 andthecyclicadjacencystructuredescribedinTheorem3.4. InrelationtoTheorem3.4,take n 0 tobesucientlylargeandpick ` sothat 6 ` >n 0 .Let H beanextremalgraphon 6 ` vertices.Weconstructtwographson 6 k + ` verticesusing H andwillrelyonthestructureof H givenbyTheorem3.4to comparethem.Construct G 1 from C k 6 byblowingupeveryvertexintoasetof 6 ` verticesandinsertacopyof H intoeach.Through ` applicationsofTheorem3.4, G 1 isanextremalgraph.Forcomparison,construct G 2 from G inthesameway.Since C G C C k 6 C G 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(C G 1 = )]TJ/F15 11.9552 Tf 5.48 9.684 Td [(6 k C H + C G ` 6 )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F15 11.9552 Tf 5.48 9.684 Td [(6 k C H + C C k 6 ` 6 = ` 6 )]TJ/F20 11.9552 Tf 5.479 9.684 Td [(C G )]TJ/F20 11.9552 Tf 11.955 0 Td [(C C k 6 0 : Thus, C G 2 C G 1 .Hence, G 2 mustalsobeextremal.Since G 2 has 6 k + ` > 6 k >n 0 vertices,Theorem3.4guaranteesthat V G 2 canbepartitionedintosets X 0 ;:::;X 5 with j X i j =6 k + ` )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 andthedescribedadjacencystructure. Wetakeamomenttoconsidersuchapartition.If u;v 2 X i ,then N u and 54
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N v bothcontain X i +1 [ X i +5 ,andneitherintersect X i +2 [ X i +3 [ X i +4 .Thusthe adjacencypattern ,thatistheneighborsandnonneighbors,of u and v agreesonmore than 3 4 of V G 2 nf u;v g ,andhasthepotentialtodisagreeonlyin X i .Therearethree casestoconsiderif u 2 X i and v 2 X j with i 6 = j .First,if u 2 X i and v 2 X i +1 ,then theadjacencypatternof u and v agreeson X i +3 [ X i +4 anddisagreeson X i +2 [ X i +5 Next,if u 2 X i and v 2 X i +2 ,thentheadjacencypatternof u and v agreeson X i +1 [ X i +4 anddisagreeson X i +3 [ X i +5 .Lastly,if u 2 X i and v 2 X i +3 ,then theadjacencypatternof u and v disagreeson X i +1 [ X i +2 [ X i +4 [ X i +5 .Therefore, if u 2 X i and v 2 X j with i 6 = j ,thentheadjacencypatternof u and v agreeon lessthan 3 4 of V G 2 nf u;v g .Therefore,apairofverticesin G 2 areinthesame X i preciselywhentheiradjacencypatternagreesonmorethan 3 4 of V G 2 nf u;v g However,considertheconstructionof G 2 .Theadjacencypatternforanypairof verticesinacopyof H agreeson G 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(H ,whichis 6 k + ` )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 ` 6 k + ` = 6 k )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 6 k 35 36 > 3 4 since k 2 .Thisimpliesthattheverticesforeachcopyof H intheblowupof G are inthesame X i ,thatis V H X i foraxedcopyof H andsome X i .Recallingthat eachvertexof G correspondstoacopyof H ,thepartitionon V G 2 inwhicheach copyof H isinthesame X i inducesapartitionof V G .Becausethispartitionon V G 2 hasthestructuredescribedinTheorem3.4,wehaveapartitionof V G with thestructuredescribedinTheorem3.4.Theexistenceofthispartitioncontradicts ourinitialchoiceof G andcompletestheproofofTheorem3.3. 55
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COMPUTATIONSINGRAPHCOLORING Graphcoloringisanimmenselypopularareaofresearchingraphtheory.Thisarea,at itscore,partitionsagraphsothattherepresentativepartssatisfyagivenproperty. Theclassicgraphcoloringproblem,vertexcoloring,partitionsthevertexsetinto independentsets.Oneofthemostcommonproblemsingraphcolorings,theFour ColorTheorem,hassignicantimpactontheworkappearinginthispart,which isjointwithJenniferDiemunschandSogolJahanbekam.Theauthorcontributeda keycomponentofthemainresultsbyprovingLemma5.11,computationallyveried coecientsofpolynomialsinLemmmas5.12,5.13,and5.14toallowapplicationofthe CombinatorialNullstellensatz,andactivelycontributedindiscoveringthedischarging rulesintegraltotheproofsinSection5.3. 56
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CHAPTER4 ANINTRODUCTIONTOGRAPHCOLORING Webeginwithsomebasicdenitions.A vertexcoloring ofagraph G isafunction : V G !C ,where C isasetofcolors.Thevertexcoloring is proper ifevery pairofadjacentverticesareassigneddistinctcolors.A k coloring isavertexcoloring thatusesatmost k colors.Agraphis k colorable ifthereexistsaproper k coloring onitsvertexset.The chromaticnumber ofagraph G ,denoted G ,isthesmallest nonnegativeinteger k suchthat G is k colorable.Inthebroadercontextofgraph coloringviewedaspartitioningtheunderlyinggraphstructure,vertexcoloringpartitionstheverticesofthegraphintoindependentsets,eachofwhichisreferredtoas a colorclass Karp[65]provedthatdeterminingthechromaticnumberofagraphisNPcomplete,whichmeansitispossibletoquicklyinpolynomialtimecheckagiven solutionbutitisnotcurrentlyknownhowtoquicklyndthesolution.Moreover,if onecouldndaquicksolution,thenalargenumberofotherproblemsbelievedtobe verydicultcouldbesolvedquicklyaswell.Althoughcomputationalapproachesto determiningthechromaticnumberofagraphmaynotbeecient,therearemany theoreticalbounds. The independencenumber ofagraph G ,denoted G ,isthesizeofamaximum independentsetin G .Thisdenitiongivesthefollowingimmediatebound: G j V G j G : Whilethisboundisrathernaive,Bollobas[15]showedittobeverycloseforalmost allgraphs. Thepresenceofedgesbothgloballyandlocallyhavebeenusedtoboundthe chromaticnumberofagraph.Fromaglobalperspective,NordhausandGaddum[71] showedforanygraph G anditsedgecomplement G G + G j V G j +1 .In 57
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alocalperspective,Brooks[23]showedthatif G isaconnectedgraphthen G G unless G iscompleteoranoddcycle. Characteristicsofvertexcoloringshavealsobeenstudied.Forexample,Hajnal andSzemerdi[51]showedthateverygraph G admitsa G +1 coloringsuch thateachcolorclasshas j j V G j G +1 k or l j V G j G +1 m vertices.Moregenerally,Kierstead andKostochka[67]showedthatif d u + d y 2 D +1 forevery xy 2 E G ,then G admitsa D +1 coloringsuchthateachcolorclasshas j j V G j D +1 k or l j V G j D +1 m vertices. 4.1TheFourColorTheoremandtheDischargingMethod Thechromaticnumberofmanyfamiliesofgraphsareknown.Forexample,a graph G haschromaticnumber G =1 ifandonlyif G hasnoedges.Agraph G haschromaticnumber G 2 ifandonlyif G isbipartite.Inparticular, trees areconnectedgraphswithnocyclesand forests aregraphsinwhicheveryconnected componentisatree.Thenicestructureoftreesmakesiteasytoshowthattrees,and subsequentlyforests,arebipartite. Additionally,forany k 2 N C 2 k =2 and C 2 k +1 =3 .The girth ofagraph isthelengthofitsshortestcycle.Itisinterestingtonotethatalthoughagraph G withlargegirthmaylookverymuchlikeatreeinsmallerareas,itispossiblefor G tohaveanarbitrarilylargechromaticnumber[32]. Anotherfamilyofgraphswithnicestructureis planar graphs,whichcanbedrawn intheplanewithnocrossingedges.Whendiscussingplanargraphs,wewilloften refertothe faces ofthegraph,whichareregionsintheplaneboundedbyaclosed walk.Foraplanegraph G ,wewilllet F G denotethesetoffacesof G .Aplane graphinwhichallfaceshavelength3iscalleda triangulation .Thenicestructureof planargraphsgivesrisetothefollowingequation: Proposition4.1Euler'sFormula[39] Foraconnected,planegraph G j V G j)222(j E G j + j F G j =2 : 58
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ThefollowingisastraightforwardapplicationofEuler'sFormula: Proposition4.2 Atriangulation G satises j E G j =3 j V G j)]TJ/F15 11.9552 Tf 17.933 0 Td [(6 Proposition4.3 Foraplanegraph G X f 2 F G l f =2 j E G j ; where l f representsthelengthofface f in G Whiletrueforallgraphs,notjustplanargraphs,weincludethefollowinghere foreaseofreference: Proposition4.4DegreeSumFormula Foranygraph G X v 2 V G d v =2 j E G j ; where d v representsthedegreeofvertex v in G Withthearbitrarilylargegappossiblebetweengirthandchromaticnumberin mind,thefollowingisperhapssomewhatsurprising. Theorem4.5TheFourColorTheorem[5,6] If G isaplanargraph,then G 4 ConjecturedbyGuthriein1852,theFourColorTheoremtookover100yearsto prove.Alongthewaymanyfalseproofswereannounced,mostnotablyofKempe [66]in1879andTait[87]in1880.Heawood[56]showedthateveryplanargraphis 5colorable,andGrtzsch[48]showedthateveryplanargraphwithgirthatleast4 is3colorable.Forasurveyofresultsoncoloringofplanegraphs,see[21].Before proceeding,itisworthmentioningboththatHadwiger'sConjecture[50]istruefor 4chromaticgraphs[31],5chromaticgraphs[94],and6chromaticgraphs[80],and alsothateachproofusestheFourColorTheorem. 59
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OneofthemostsignicantoutcomesfromeortstoprovetheFourColorTheorem isthedevelopmentoftheDischargingMethod.Thegeneralapproachtothismethod canbroadlybedescribedintwosteps.First,globalpropertiesareusedtoguarantee theexistenceoflocalcongurations,whichareoftenreferredtoasunavoidable congurations.Then,theselocalcongurationsareshownnottoexistinaminimal counterexampletothedesiredconclusion,thatisthecongurationsarereducible. Thus,thegoalofmanyproofsistondalistofunavoidablecongurationsandshow thatallofthosecongurationsarereducible. Provingthatunavoidablecongurationsarereducibleofteninvolvesdetailedcase analysis.Insomeinstances,thecaseanalysisinvolvediswellsuitedforcomputerassistance.Forexample,AppelandHaken'sproofoftheFourColorTheorem[5,6] usedcomputerstohelpconrmthereducibilityof1,936congurationsunder487dischargingrules.Widelytoutedastherstcomputerassistedmathematicalproof,the validityoftheresulthaswithstoodmuchskepticism.In1997,Robertson,Sanders, Seymour,andThomas[79]presentedaproofusingonly633reduciblecongurations and32dischargingrules.Althoughneedingsignicantlyfewerreduciblecongurations,theproofstillutilizedcomputerassistance.FurtherexpositionabouttheFour ColorTheoremanditsproofcanbefoundinanarticlebyWoodall[96]andabook byWilson[95]. Todiscoverunavoidablecongurations,chargeisassignedtoelementsofagraph andthenreallocatedaroundthegraphunderasetofrules,thatisdischargingrules. Todiscoverunavoidablecongurationsinthefollowingexamplethatappearsin[19], weneedthefollowingdenition:a k edgeconnectedgraphis essentially k +1 edgeconnected ifallofits k edgecutsareassociatedwithasinglevertex.Asanimmediate consequenceofthisdenition,theminimumdegreeofanessentially k edgeconnected graphis k )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Proposition4.6 Everyessentially6connectedtriangulation G mustcontainaver60
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texofdegree5adjacenttoavertexofdegree5or6. Proof: Assigneachvertex v achargeof 6 )]TJ/F20 11.9552 Tf 11.977 0 Td [(d v ,where d v representsthedegree of v .FromPropositions4.4and4.2,thetotalchargeofthegraphis X v 2 V G )]TJ/F15 11.9552 Tf 5.48 9.684 Td [(6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(d v =6 j V G j)]TJ/F26 11.9552 Tf 24.325 11.358 Td [(X v 2 V G d v =6 j V G j)]TJ/F15 11.9552 Tf 17.933 0 Td [(2 j E G j =12 : Ouraimindischargingistoreallocatechargesothateveryvertexhasnonpositive charge,whichisimpossibleasthetotalchargeis12andnochargeisdestroyed.Notice thatverticesofdegreeatleast6aregivenanonpositiveinitialcharge.Since G isan essentially6connectedtriangulation,ithasminimumdegree5.Thus,weneedonly distributechargeawayfromverticesofdegree5withoutgivingawaysomuchcharge thatotherverticesendupwithpositivecharge. Considerthefollowingreallocationofcharge:foreveryvertexofdegree5,distributeits1chargeequallyamongits5neighbors.Everyvertex v canreceiveatmost 1 5 d v .Thus,avertexofdegreeatleast8hasanalchargeof 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(d v + 1 5 d v =6 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 5 d v 6 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(32 5 < 0 : Similarly,eachvertexofdegree7withatmost5neighborsofdegree5hasnonpositive nalcharge,asdoverticesofdegree5or6withnoneighborofdegree5.Thus, G musthaveeitheravertexofdegree5adjacenttoavertexofdegree5or6,oravertex ofdegree7with6or7neighborsofdegree5. However,since G isatriangulation,consecutiveneighborsofavertexareadjacent. Thus,theneighborhoodofavertexofdegree7withatleast4neighborsofdegree5 hasapairofadjacentverticesofdegree5.Hence,itisenoughtosaythat G must haveeitheravertexofdegree5adjacenttoavertexofdegree5or6. Dischargingargumentsappearinresultsrelatedtoa1975conjectureofSteinberg, whichstatesthatplanargraphswithout4or5cyclesare3colorable.Notethatthis conjecturehasveryrecentlybeendisprovedbyCohenAdda,Hebdige,Krl,Li,and 61
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Salgado[27].AparticularlynicedischargingargumentappearsinAbbottandZhou [1]regardingthe3colorabilityofplanargraphswithno k cyclesfor 4 k 11 ,which wasimprovedbyBorodin,Glebov,Raspaud,andSalavatipour[22]to 4 k 7 CranstonandWest[28]provideafriendly,comprehensiveguidetothedischarging methodthatincludesavarietyofexamplesandexistingresults.Salavatipour[81] providesanicesurveyofdischargingresultsinhisdissertation. Ourresultscenteraroundaspeciccoloringofplanargraphsandusecomputer assistancetoprovethatcertaincongurationsarereducible.Ratherthanchecking longlistsofdetailedcasestodoso,weembedinformationrelevanttoourcoloringinto polynomialsandthencombinecomputingpowerwiththetheoreticalresultpresented inthefollowingsection. 4.2ListColoringandtheCombinatorialNullstellensatz Anaturalgeneralizationofvertexcoloringis listcoloring .Whileavertexcoloring colorseachvertexfromasinglepalletofcolors,everyvertexiscoloredfromitsown palletofcolorsinalistcoloring.Moreformally,a listassignment onagraph G isafunction L : V G 2 R .Agraph G is L colorableor listcolorable whenthe contextisclearif G admitsapropervertexcoloring suchthat v 2 L v forall v 2 V G .Agraphis k choosable ifitis L colorableforeverylistassignment L with j L v j k forall v 2 V G .The listchromaticnumber of G ,denoted ` G ,isthe smallest k 2 N suchthat G is k choosable. Thereisavastcollectionofworkrootedincomparingthelistchromaticnumber ofagraphtoitschromaticnumber.Tomentionafew,Erds,Rubin,andTaylor [36]showedthatthesetwographparameterscanbearbitrarilyfarapart.Forany graph G ,theyalsoshowedthat ` G + ` G j V G j +1 and ` G G ,the upperboundsofwhichmatchupperboundsfor G .InrelationtotheFourColor Theorem,Thomassen[88]showedthateveryplanargraphis5choosableandVoigt [92]showedthatthereexistnon4choosableplanargraphs.InrelationtoGrtzsch's 62
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Theorem[48],KratochvlandTuza[68]showedthateveryplanargraphofgirthat leastfouris4choosable,Thomassen[89]showedthateveryplanargraphofgirth atleastveis3choosable,andVoigt[93]showedthatthereexistnon3choosable planargraphswithgirthatleastfour. AusefultoolforlistcoloringandlistversionsofothergraphcoloringsisAlon's CombinatorialNullstellensatz.Applicationofthisusefultheoremhasoccurredin mathematicaleldsotherthangraphtheoryandcombinatoricssuchasadditive numbertheoryanddiscretegeometry.Formallystatedbelow,theCombinatorial Nullstellensatzprovidescriteriatoguaranteetheexistenceofanonzerosolutiontoa multivariatepolynomialfromacollectionofsubsetsofaeld. Theorem4.7CombinatorialNullstellensatz[3] If Q n i =1 x t i i isamonomial withnonzerocoecientinapolynomial f havingdegree P n i =1 t i overaeld F ,and S 1 ;:::;S n aresetswith j S i j >t i for i 2 [ n ] ,then f x 6 =0 forsome x 2 Q n i =1 S i InapplyingtheCombinatorialNullstellensatztoreducibilityargumentswithin dischargingproofs,weconstructcoloringpolynomialswhosezeroscorrespondto breakingtherulesofthedesiredcoloring.Aftercomputationallyverifyingtheexistenceofamonomialhavingmaximumdegreeinthesecoloringpolynomials,wecan applytheCombinatorialNullstellensatztoobtainanonzerosolution.Anonzero solutiontoacoloringpolynomialcorrespondstoapropercoloringofthegraphcongurationusedtoconstructthecoloringpolynomial.Thesegraphcongurationsare infactthereduciblecongurationsforourresults. Workingwithinaminimalcounterexamplesupposedlycontainingsuchaconguration,wedeleteportionsoftheconguration,inductivelycolortheremaining graph,andthenconstructacoloringpolynomialthatreectstherestrictionsofextendingtheinductivecoloringtotheentireminimalcounterexample.Application oftheCombinatorialNullstellensatztotheconstructedcoloringpolynomialguaran63
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teestheexistenceofapropercoloringontheminimalcounterexamplecontainingthe conguration,whichmakesthecongurationreduciblesincetheminimalcounterexamplecannotbeproperlycolored.Thisapplicationispreciselywhereourarguments avoidmuchofthedetailedcaseanalysisoftenpresentwithinreducibilityarguments fordischarging. 64
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CHAPTER5 LUCKYLABELING Thereareamultitudeofvariationsonvertexcoloring,seeGallian'sdynamic survey[46].Weconsideraderivedvertexcoloringinwhicheachvertexreceivesa colorbasedonassignedlabelsofitsneighbors.Let ` : V G R bealabelingofthe verticesofagraph G .Foreach v 2 V G ,let S G v = P u 2 N G v ` u ,where N G v is theopenneighborhoodof v in G .Whenthecontextisclear,weuse S v inplaceof S G v .Wesaythelabeling ` is lucky ifforeverypairofadjacentvertices u and v ,we have S u 6 = S v ,thatis S inducesapropervertexcoloringof G .Theleastinteger k forwhichagraph G hasaluckylabelingusinglabelsfrom f 1 ;:::;k g iscalledthe luckynumber of G ,denoted G Determiningtheluckynumberofagraphisanaturalvariationofawellstudied problemposedbyKaroski,uczak,andThomason[64],inwhichedgelabelsfrom f 1 ;:::;k g aresummedatincidentverticestoinduceavertexcoloring.Karoski et al. conjecturethatedgelabelsfrom f 1 ; 2 ; 3 g areenoughtoyieldapropervertex coloringofgraphswithnocomponentisomorphicto K 2 .Thisconjectureisknown asthe 1 2 3 Conjectureandisstillopen.In2010,Kalkowski,Karoski,andPfender [63]showedthatlabelsfrom f 1 ; 2 ; 3 ; 4 ; 5 g suce. Similartotheluckynumberofagraph,Chartrand,Okamoto,andZhang[25] dened G tobethesmallestinteger k suchthat G hasaluckylabelingusing k distinctlabels.Theyshowedthat G G .Notethat G G ,sincewith G weseekthesmallest k suchthatlabelsarefrom f 1 ;:::;k g whileallowingsome integersin f 1 ;:::;k g tonotbeusedaslabels,whereas G considersthefewest distinctlabelswithoutconcernforthevalueofthelargestlabel. In2009,Czerwiski,Grytczuk,andelaznyproposedthefollowingconjecturefor theluckynumberof G : 65
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Conjecture5.1Czerwiski,Grytczuk,elazny[29] Foreverygraph G G G Thisconjectureremainsopenevenforbipartitegraphs,forwhichnoconstant boundiscurrentlyknown.Czerwiski etal. [29]showedthat G k +1 forevery bipartitegraph G havinganorientationinwhicheachvertexhasoutdegreeatmost k .Theyalsoshowedthat G 2 when G isatree, G 3 when G isbipartite andplanar,and G 100 ; 280 ; 245 ; 065 foreveryplanargraph G .Notethatifthe conjectureistrue,then G 4 foranyplanargraph G .Theboundforplanar graphswaslaterimprovedto G 468 byBartnicki etal. [13],whoalsoshowthe following. Theorem5.2Bartnickietal.[13] If G isa 3 colorableplanargraph,then G 36 Thegirthofagraphisespeciallyusefulingivingameasureofsparseness.Knowingthegirthofaplanargraphgivesaboundonthe maximumaveragedegree ofa graph G ,denoted mad G ,whichisthemaximumaveragedegreeoverallsubgraphs of G .ThefollowingpropositionisasimpleapplicationofEuler'sformulaProposition 4.1andgivesarelationshipbetweenthesetwoparameters: Proposition5.3 If G isaplanargraphwithgirth g ,then mad G < 2 g g )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 Proof: Let H G .Notethatthegirthof H isatleast g .Beginningwiththe averagedegreeof H andapplyingEuler'sformulaProposition4.1,wehave P v 2 V H d v j V H j = 2 j E H j 2+ j E H j)222(j F H j < 2 j E H j j E H j)222(j F H j : Notethateachfacehasatleast g edges.Becauseeachedgeiscountedexactlytwice insummingthelengthsoffacesinaplanargraph, g j F H j 2 j E H j .Thus, P v 2 V H j V H j < g g 2 j E H j j E H j)222(j F H j 2 g j E H j g j E H j)]TJ/F15 11.9552 Tf 17.933 0 Td [(2 j E H j = 2 g g )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 : 66
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Bythearbitrarychoiceof H ,everysubgraphof G satisestheaboveinequality,which givesthedesiredresult. Bartnickietal.[13]provedthefollowing: Theorem5.4Bartnickietal.[13] If G isaplanargraphofgirthatleast 13 then G 4 In2013,Akbari,Ghanbari,Manaviyat,andZare[2]proposedthelistversionof luckylabeling.Agraphis luckykchoosable ifwhenevereachvertexisgivenalistof atleast k availableintegers,aluckylabelingcanbechosenfromthelists.The lucky choicenumber ofagraph G istheminimumpositiveinteger k suchthat G islucky k choosable,andisdenotedby ` G .Akbari etal. [2]showedthat ` G 2 )]TJ/F15 11.9552 Tf 10.943 0 Td [(+1 foreverygraph G with 2 .Theyalsoprovedthefollowing: Theorem5.5Akbarietal.[2] If G isaforest,then ` G 3 Inthispaper,weimprovetheseresultsforplanargraphsofparticulargirths. Specically,weusetheCombinatorialNullstellensatzwithinreducibilityarguments ofthedischargingmethodtoproveourresults.Thecombinationofthesetwopopular techniquesisanovelapproachthatcaneliminateaconsiderableamountofcase analysis.Moreover,usingtheCombinatorialNullstellensatzinreducibilityarguments ofcoloringproblemsenablesprovingchoosabilityresults,ratherthanjustcolorability. Weshowthefollowingimprovementsontheluckychoicenumberforplanargraphs ofgivengirths: Theorem5.6 Let G beaplanargraphwithgirth g 1.If g 5 ,then ` G 19 2.If g 6 ,then ` G 9 3.If g 7 ,then ` G 8 67
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4.If g 26 ,then ` G 3 Various3coloringsofplanargraphshavebeenobtainedundercertaingirthassumptions.CombinedwithGrtzsch'sresult[48],ourresultanswersConjecture5.1 fornonbipartiteplanargraphswithgirthatleast26. InSection5.1,weintroducethenotationandtoolsthatareusedthroughoutthe remainderofthechapter.Wealsogiveanoverviewofhowweusethedischarging methodandtheCombinatorialNullstellensatz.Section5.2describescertainreducible congurations.Finally,inSection5.3weproveTheorem5.6. 5.1NotationandTools Let N G v betheopenneighborhoodofavertex v inagraph G .Forconvenience, a j vertex j )]TJ/F38 11.9552 Tf 7.085 4.338 Td [(vertex ,or j + vertex isavertexwithdegree j ,atmost j ,oratleast j ,respectively.Similarly,a j neighbor respectively j )]TJ/F38 11.9552 Tf 7.084 4.338 Td [(neighbor or j + neighbor of v isa j vertexrespectively j )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(vertexor j + vertexadjacentto v Forsets A and B ofrealnumbers, A B isdenedtobetheset f a + b : a 2 A;b 2 B g .Likewise, A B isdenedtobetheset f a )]TJ/F20 11.9552 Tf 12.348 0 Td [(b : a 2 A;b 2 B g .When B = ; ,wedene A B = A B = A .Weusethefollowingknownresultfrom additivecombinatorics: Proposition5.7 Let A 1 ;:::;A r benitesetsofrealnumbers.Wehave j A 1 A r j 1+ r X i =1 j A i j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 : Proof: Weapplyinductionon r P i =1 j A i j .When r P i =1 j A i j =1 ,allbutone A i areempty, sowehave j A 1 A r j =1 ,asdesired. Now,supposethat r P i =1 j A i j = n .Wemaysupposethatall A i arenonempty. Let a i betheminimumelementof A i for i 2f 1 ;:::;n g .Let A 0 1 = A 1 )261(f a 1 g .By theinductionhypothesis,wehave j A 0 1 A 2 A r j 1+ r P i =1 j A i j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 )]TJ/F15 11.9552 Tf 13.097 0 Td [(1 However, j A 1 A r jjf a 1 + + a r gj + j A 0 1 A 2 A r j .Therefore, j A 1 A r j 1+ r P i =1 j A i j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1 68
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Notethat A )]TJ/F20 11.9552 Tf 9.298 0 Td [(B isthesameas A B ,where )]TJ/F20 11.9552 Tf 9.299 0 Td [(B = f)]TJ/F20 11.9552 Tf 15.276 0 Td [(b : b 2 B g .Thisyields thefollowingknowncorollary: Corollary5.8 Let A and B benonemptysetsofpositiverealnumbers.Wehave j A B jj A j + j B j)]TJ/F15 11.9552 Tf 17.932 0 Td [(1 Throughout,weconsiderwhenendpointsofedgesneeddierentsumstoyielda luckylabeling.Forthisreason,ifweknow S u 6 = S v foranedge uv of G ,thenwe saythat uv is satised ; uv is unsatised otherwise. Ourproofsrelyonapplyingthedischargingmethod.Thisprooftechniqueassigns aninitialchargetoverticesandpossiblyfacesofagraphandthendistributescharge accordingtoalistofdischargingrules.Acongurationis k reducible ifitcannot occurinavertexminimalgraph G with ` G >k .Notethatany k reducible congurationisalso k +1 reducible.Whenapplyingthedischargingmethodin Theorem5.19,werequirethefollowingknownlemma,whichisasimpleapplication ofEuler'sFormula: Proposition5.9 Givenaplanargraph G X f 2 F G l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(4+ X v 2 V G d v )]TJ/F15 11.9552 Tf 11.955 0 Td [(4= )]TJ/F15 11.9552 Tf 9.299 0 Td [(8 : Proof: Let G beaplanargraph.ByEuler'sFormulaProposition4.1,wehave that 2= j V G j)222(j E G j + j F G j : Multiplyingby4gives )]TJ/F15 11.9552 Tf 9.298 0 Td [(8=2 j E G j)]TJ/F15 11.9552 Tf 17.932 0 Td [(4 j V G j +2 j E G j)]TJ/F15 11.9552 Tf 17.933 0 Td [(4 j F G j : ApplyingPropositions4.4and4.3andgroupingtermsgivesthedesiredresult. Wealsorequirealargeindependentset,whichisgivenfromthefollowingtheorem: 69
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Theorem5.10Steinberg,Tovey[86] Everyplanartrianglefreegraphon n verticeshasanindependentsetofsizeatleast n +1 3 Themaintoolweusetodeterminewhencongurationsare k reducibleisthe CombinatorialNullstellensatz,whichisappliedtocertaingraphcongurations. 5.2ReducibleCongurations Inthelemmasinthissection,welet k 2 N andintroduce k reduciblecongurationsthatwillbeusedtoproveourmainresult.Let L : V G 2 R beafunctionon V G suchthat jL v j = k foreach v 2 V G .Thus, L v denotesalistof k available labelsfor v .Ineachproof,wetake G tobeavertexminimalgraphwith ` G >k Then,wedeneapropersubgraph G 0 of G with V G 0 V G .Bythechoiceof G G 0 hasaluckylabeling ` suchthat ` v 2L v forall v 2 V G 0 .Thislabelingof G 0 isthenextendedtoaluckylabelingof G bydening ` v for v 2 V G )]TJ/F20 11.9552 Tf 11.703 0 Td [(V G 0 .We discussthedetailsofthisapproachinLemmas5.11and5.12.Theremaininglemmas aresimilarinapproach,soweincludefewerdetailsintheproofs. Lemma5.11 Thefollowingcongurationsare k reducibleintheclassofgraphswith girthatleast5: aavertex v with P u 2 N v d u d v Proof: Assume G isavertexminimalgraphwith ` G >k containingthe congurationdescribedina.Let G 0 = G )97(f v g .Since G isvertexminimal, ` G 0 k .Let ` bealuckylabelingfrom L on V G 0 .Ouraimistochoose ` v from L v to extendtheluckylabelingof G 0 toaluckylabelingof G .Notethattheonlyunsatised edgesof G arethoseincidenttoneighborsof v .Let e beanedgeincidenttoaneighbor 70
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u of v .If e = uv ,then e issatisedwhen ` v 6 = P w 2 N v ` w )]TJ/F20 11.9552 Tf 12.504 0 Td [(S G 0 u .If e = uw forsome w 6 = v ,then e issatisedwhen ` v 6 = S G 0 w )]TJ/F20 11.9552 Tf 12.083 0 Td [(S G 0 u .Thus,picking ` v distinctfromatmost P u 2 N v d u valuesensuresthatalledgesof G aresatised.Since P u 2 N v d u k containingthecongurationdescribedinb.Let R bethesetof r 1 neighborsof v .Let G 0 = G )]TJ/F15 11.9552 Tf 12.695 0 Td [( R [ Q .Since girth G 5 Q isindependent.Therefore,for each i 2f 1 ;:::;q g ,thereareatleast jL v i j)]TJ/F20 11.9552 Tf 19.135 0 Td [(d G v 0 i choicesfor ` v i thatensure alledgesincidentto v 0 i aresatisedin G .Consider vw in E G .If w 2 V G 0 vw issatisedwhen P x 2 R [ Q ` x 6 = S G 0 w )]TJ/F20 11.9552 Tf 12.255 0 Td [(S G 0 v .Also,if w 2 R ,then vw issatised when P x 2 R [ Q ` x 6 = ` v )]TJ/F20 11.9552 Tf 11.012 0 Td [(S G 0 v .If w = v i forsome v i 2 Q ,then vw issatisedwhen P x 2 R [ Q ` x 6 = ` v + ` w )]TJ/F20 11.9552 Tf 12.767 0 Td [(S G 0 v .Therefore,wemustavoidatmost d v values for P x 2 R [ Q ` x inordertosatisfyalledgesincidentto v .Recallthateachvertex in R and Q have k and k )]TJ/F20 11.9552 Tf 12.912 0 Td [(d v 0 i labels,respectively,thatavoidrestrictedsums. Proposition5.7guaranteesatleast 1+ r k )]TJ/F15 11.9552 Tf 11.491 0 Td [(1+ P v i 2 Q 0 k )]TJ/F20 11.9552 Tf 11.49 0 Td [(d v 0 i )]TJ/F15 11.9552 Tf 11.49 0 Td [(1 availablevalues for P w 2 R ` w + P w 2 Q ` w .Since,byassumption, 1+ r k )]TJ/F15 11.9552 Tf 10.416 0 Td [(1+ P v i 2 Q k )]TJ/F20 11.9552 Tf 10.416 0 Td [(d v 0 i )]TJ/F15 11.9552 Tf 10.416 0 Td [(1 >d v thereisatleastonechoicefor ` w foreach w in R [ Q thatcompletesaluckylabeling of G .Thus, ` G k ,acontradiction. Lemma5.12 Thefollowingcongurationsare 8 reducibleintheclassofgraphsof girthatleast6: aa 6 vertex v havingsix 2 neighborsoneofwhichhasa 3 )]TJ/F38 11.9552 Tf 7.085 4.339 Td [(neighbor,and ba 7 vertex v havingseven 2 neighborstwoofwhichhave 4 )]TJ/F38 11.9552 Tf 7.085 4.338 Td [(neighbors. Proof: Let G beavertexminimalgraphofgirthatleast6with ` G > 8 Tothecontrary,suppose G containsthecongurationdescribedina.Let u bea 71
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2 neighborof v havinga 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.Let G 0 = G )219(f u;v g .Let ` : V G 0 R bea luckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G Theonlyunsatisededgesof G arethoseincidenttoneighborsof u or v .To satisfytheunsatisededgesnotincidentto u or v ,weavoidatmosttwovaluesfrom L u andatmostvevaluesfrom L v .Notethat jL u j 8 and jL v j 8 .Thus, thereareatleastsixlabelsavailablefor u andatleastthreeavailablefor v .Tosatisfy theedgesincidentto u or v ` u )]TJ/F20 11.9552 Tf 12.071 0 Td [(` v mustavoidatmostsevenvalues.Corollary 5.8givesatleasteightvaluesfor ` u )]TJ/F20 11.9552 Tf 12.375 0 Td [(` v fromavailablelabels.Thus,thereare labelsthatcompletealuckylabelingof G .Hence, ` G 8 ,acontradiction. v u 1 u 0 1 u 2 u 0 2 u 3 u 0 3 u 4 u 0 4 u 5 u 0 5 u 6 u 0 6 u 7 u 0 7 w 1 w 2 w 3 w 0 1 w 0 2 w 0 3 Figure5.1:An8reducibleconguration. Now,weprovepartb.Tothecontrarysuppose G containstheconguration describedinb.Let u 1 ;:::;u 7 bethe2neighborsof v whoseotherneighborsare u 0 1 ;:::;u 0 7 ,respectively,where u 0 1 and u 0 2 are 4 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(vertices.Sincethemostrestrictions onlabelsoccurswhen d u 0 1 = d u 0 2 =4 ,weassumethisisthecase.Let N u 0 1 )]TJ 422.702 23.98 Td [(f u 1 g = f w 1 ;w 2 ;w 3 g and N u 0 2 )295(f u 2 g = f w 0 1 ;w 0 2 ;w 0 3 g seeFigure5.1.Consider G 0 = G )332(f v;u 1 ;u 2 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .Theonlyunsatisededgesof G arethoseincident to u 0 1 u 0 2 ,andneighborsof v .Thefollowingfunctionhasfactorsthatcorrespondto unsatisededges,where x y ,and z representthepossiblevaluesof ` v ` u 1 ,and 72
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` u 2 ,respectively: f x;y;z = 7 Y i =1 y + z + 7 X j =3 ` u j )]TJ/F20 11.9552 Tf 11.955 0 Td [(x )]TJ/F20 11.9552 Tf 11.955 0 Td [(` u 0 i 7 Y i =3 x + ` u 0 i )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 u 0 i 3 Y i =1 y + S G 0 u 0 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 w i 3 Y i =1 z + S G 0 u 0 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(S G 0 w 0 i x + ` u 0 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(y )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 u 0 1 x + ` u 0 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(z )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 u 0 2 : Thecoecientof x 7 y 6 z 7 in f x;y;z isequaltoitscoecientin y + z )]TJ/F20 11.9552 Tf 10.199 0 Td [(x 7 x 5 y 3 z 3 x )]TJ/F20 11.9552 Tf 422.702 23.98 Td [(y x )]TJ/F20 11.9552 Tf 10.167 0 Td [(z ,whichis490.TheCombinatorialNullstellensatzTheorem4.7impliesthat thereisachoiceoflabelsfor ` v ` u 1 ,and ` u 2 fromlistsofsizeatleast8that make f nonzero.Thus,theselabelsinducealuckylabelingof G .Hence, ` G 8 acontradiction. Lemma5.13 Acongurationthatisaninducedcyclewithvertices v 1 v 2 v 3 v 4 v 5 such that d v 1 17 d v 2 = d v 5 =2 d v 3 7 ,and d v 4 7 is19reducible. v 1 v 1 v 1 v 2 v 2 v 2 v 3 v 3 v 3 v 4 v 4 v 4 v 5 v 5 v 5 Figure5.2:Areducibleconguration. Proof: Let G beavertexminimalgraphwith ` G > 19 .Supposetothe contrarythat G containsthecongurationinFigure5.2.Sincethemostrestrictions onlabelsoccurswhen d v 1 =17 and d v 3 = d v 4 =7 ,weassumethisisthe case.Let G 0 = G )240(f v 2 ;v 5 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .Theunsatisededgesarethoseincidentto v 1 ;:::;v 5 Thefollowingfunctionhasfactorscorrespondingtotheunsatisededgeswhere x 2 73
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and x 5 representlabelsof v 2 and v 5 ,respectively: f x 2 ;x 5 = S G 0 v 1 + x 2 + x 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 3 ` v 1 + ` v 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 3 x 2 + S G 0 v 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 4 x 5 + S G 0 v 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 4 ` v 1 + ` v 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 1 Y w 2 N G 0 v 4 f v 3 g S G 0 w )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 5 Y w 2 N G 0 v 1 S G 0 w )]TJ/F20 11.9552 Tf 11.956 0 Td [(S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 5 Y w 2 N G 0 v 3 f v 4 g S G 0 w )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 2 : Thecoecientof x 16 2 x 14 5 in f x 2 ;x 5 isthesameas x 10 2 x 8 5 in )]TJ/F15 11.9552 Tf 9.299 0 Td [( x 2 + x 5 17 x 2 )]TJ/F20 11.9552 Tf 12.356 0 Td [(x 5 whichis )]TJ/F18 7.9701 Tf 5.48 4.379 Td [(17 10 )]TJ/F26 11.9552 Tf 12.324 9.684 Td [()]TJ/F18 7.9701 Tf 5.479 4.379 Td [(17 9 .TheCombinatorialNullstellensatzTheorem4.7impliesthat ` G 19 ,acontradiction. Lemma5.14 Let P t 2 ;:::;t n )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 bethepath v 1 v n suchthatforeach i 2f 2 ;:::;n )]TJ/F15 11.9552 Tf 426.032 23.981 Td [(1 g thevertex v i has t i 1neighborsand d v i =2+ t i .Thecongurations P ; 0 ; 1 P ; 1 ; 1 P ; 1 ; 0 ; 0 P ; 1 ; 0 ; 0 P ; 0 ; 0 ; 0 ,and P ; 0 ; 0 ; 0 ; 0 are3reducible. v 1 v 2 v 3 v 4 v 5 v 6 v 7 a v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 b v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 c v 1 v 2 v 3 v 4 v 5 v 6 v 7 d v 1 v 2 v 3 v 4 v 5 v 6 v 7 e v 1 v 2 v 3 v 4 v 5 v 6 v 7 f Figure5.3:Some3reduciblecongurations. Proof: Fortheentiretyofthisproof,let G beavertexminimalgraphwith ` G > 3 74
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Supposetothecontrarythat G contains P ; 0 ; 1 ,seeFigure5.3a.Let G 0 = G )124(f v 3 ;v 6 ;v 7 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v for each v 2 V G .Theunsatisededgesarethoseincidentto v 2 ;v 3 ;v 4 .Thefollowing functionhasfactorscorrespondingtotheunsatisededgeswhere x 3 x 6 ,and x 7 representlabelsof v 3 v 6 ,and v 7 ,respectively: f x 3 ;x 6 ;x 7 = S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 6 ` v 1 + x 3 + x 6 )]TJ/F20 11.9552 Tf 11.956 0 Td [(` v 2 ` v 1 + x 3 + x 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 4 ` v 2 + ` v 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 ` v 5 + x 3 + x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 4 ` v 5 + x 3 + x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 5 : Thecoecientof x 2 3 x 2 6 x 2 7 in f is9.TheCombinatorialNullstellensatzTheorem 4.7impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 0 ; 1 is3reducible. Supposetothecontrarythat G contains P ; 1 ; 1 ,seeFigure5.3b.Let G 0 = G )144(f v 3 ;v 6 ;v 7 ;v 8 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .Theunsatisededgesarethoseincidentto v 2 ;v 3 ;v 4 .Thefollowing functionhasfactorscorrespondingtotheunsatisededgeswhere x 3 x 6 x 7 ,and x 8 representlabelsof v 3 v 6 v 7 ,and v 8 ,respectively: f x 3 ;x 6 ;x 7 ;x 8 = S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 6 ` v 1 + x 3 + x 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 2 ` v 1 + x 3 + x 6 )]TJ/F20 11.9552 Tf 11.956 0 Td [(` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 4 ` v 2 + ` v 4 + x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 ` v 2 + ` v 4 + x 7 )]TJ/F20 11.9552 Tf 11.956 0 Td [(` v 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 8 ` v 5 + x 3 + x 8 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 4 ` v 5 + x 3 + x 8 )]TJ/F20 11.9552 Tf 11.956 0 Td [(S G 0 v 5 : Thecoecientof x 2 3 x 2 6 x 7 x 2 8 is15.TheCombinatorialNullstellensatzTheorem4.7 impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 0 ; 1 is3reducible. Suppose G contains P ; 1 ; 0 ; 0 ,seeFigure5.3c.Let G 0 = G )76(f v 3 ;v 4 ;v 7 ;v 8 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .The 75
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unsatisededgesarethoseincidentto v 2 ;:::;v 5 .Thefollowingfunctionhasfactors correspondingtotheunsatisededgeswhere x 3 x 4 x 7 ,and x 8 representlabelsof v 3 v 4 v 7 ,and v 8 ,respectively: f x 3 ;x 4 ;x 7 ;x 8 = x 3 + x 7 + ` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 1 x 3 + x 7 + ` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 2 x 4 + x 8 + ` v 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 x 4 + x 8 + ` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 x 4 + x 8 + ` v 2 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 5 x 3 + ` v 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 6 x 4 + ` v 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 6 : Thecoecientof x 3 x 2 4 x 2 7 x 2 8 is8.TheCombinatorialNullstellensatzTheorem4.7 impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 1 ; 0 ; 0 is3reducible. Suppose G contains P ; 1 ; 0 ; 0 ,seeFigure5.3d.Let G 0 = G )236(f v 3 ;v 4 ;v 7 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .The unsatisededgesarethoseincidentto v 2 ;:::;v 5 .Thefollowingfunctionhasfactors correspondingtotheunsatisededgeswhere x 3 x 4 ,and x 7 representlabelsof v 3 v 4 and v 7 ,respectively: f x 3 ;x 4 ;x 7 = S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 ` v 1 + x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 4 ` v 2 + x 7 + x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 ` v 2 + x 7 + x 4 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 5 x 3 + ` v 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 6 x 4 + ` v 6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 6 : Thecoecientof x 2 3 x 2 4 x 2 7 is6.TheCombinatorialNullstellensatzTheorem4.7 impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 1 ; 0 ; 0 is3reducible. Suppose G contains P ; 0 ; 0 ; 0 ,seeFigure5.3e.Let G 0 = G )240(f v 3 ;v 4 ;v 7 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .The unsatisededgesarethoseincidentto v 2 ;:::;v 5 .Thefollowingfunctionhasfactors correspondingtotheunsatisededgeswhere x 3 x 4 ,and x 7 representlabelsof v 3 v 4 76
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and v 7 ,respectively: f x 3 ;x 4 ;x 7 = S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 ` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 7 ` v 2 + x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 7 ` v 2 + x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 5 ` v 6 + x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 5 ` v 6 + x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S G 0 v 6 : Thecoecientof x 2 3 x 2 4 x 2 7 is7.TheCombinatorialNullstellensatzTheorem4.7 impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 0 ; 0 ; 0 is3reducible. Suppose G contains P ; 0 ; 0 ; 0 ; 0 ,seeFigure5.3f.Let G 0 = G )167(f v 3 ;v 4 ;v 5 g .Let ` : V G 0 R bealuckylabelingof G 0 suchthat ` v 2L v foreach v 2 V G .The unsatisededgesarethoseincidentto v 2 ;:::;v 6 .Thefollowingfunctionhasfactors correspondingtotheunsatisededgeswhere x 3 x 4 ,and x 5 representlabelsof v 3 v 4 and v 5 ,respectively: f x 3 ;x 4 ;x 5 = S G 0 v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 3 ` v 1 + x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 4 ` v 2 + x 4 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 3 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 5 x 3 + x 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(x 4 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 6 x 4 + ` v 6 )]TJ/F20 11.9552 Tf 11.956 0 Td [(x 5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(` v 7 x 5 + ` v 7 )]TJ/F20 11.9552 Tf 11.956 0 Td [(S G 0 v 7 : Thecoecientof x 2 3 x 2 4 x 2 5 is7.TheCombinatorialNullstellensatzTheorem4.7 impliesthat ` canbeextendedtoaluckylabelingof G ,acontradiction.Thus, P ; 0 ; 0 ; 0 ; 0 is3reducible. 5.3LuckyLabelingResults Inthissection,weprovethefourresultsenumeratedinTheorem5.6. Theorem5.15 If G isaplanargraphwith girth G 5 ,then ` G 19 Proof: Let G beaplanargraphwithgirthatleast 5 andsupposethat G is vertexminimalwith ` G > 19 .ByProposition5.3, mad G < 10 = 3 .Assigneach vertex v aninitialcharge d v ,andapplythefollowingdischargingrules: R1each 1 vertexreceives 7 = 3 chargefromitsneighbor; 77
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R2each 2 vertex awithtwo 8 + neighborsreceives 2 = 3 chargefromeachneighbor, bwitha 4 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(neighboranda 15 + neighborreceives 4 = 3 chargefromits 15 + neighbor,or cwitha 10 + neighborandaneighborofdegree 5 6 ,or 7 receives 1 charge fromits 10 + neighborand 1 = 3 chargefromitsotherneighbor;and R3each 3 vertexreceives 1 = 3 chargefroma 6 + neighbor. Acontradictionwith mad G < 10 = 3 occursifthedischargingrulesreallocate chargesothateveryvertexhasnalchargeatleast 10 = 3 ;weshowthatthisisthe case. ByLemma5.11a,each 1 vertexhasa 19 + neighbor, 2 verticeshaveneighbors withdegreesumatleast 19 ,and 3 verticeshaveatleastone 6 + neighbor.Thus, bythedischargingrules, 3 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(verticeshavenalcharge 10 = 3 .Since 4 verticesneither givenorreceivecharge,theyhavenalcharge 4 Verticesofdegree d with d 2f 5 ; 6 ; 7 g givechargewhenincidentto 3 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(vertices. Bythedischargingrules,theygiveawayatmost d= 3 charge.Thisresultsinanal chargeofatleast d )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(d 3 = 2 d 3 10 3 ,since d 5 Weconsiderverticesofdegree d with d 2f 8 ; 9 g .ByLemma5.11a,each 9 vertexhasatleastone 3 + neighbor.Also,each 8 vertexhasatleasttwo 3 + neighborsoratleastone 4 + neighbor.Bythedischargingrules,thenalchargeof any 9 vertexisatleast 9 )]TJ/F15 11.9552 Tf 12.133 0 Td [(8 2 3 )]TJ/F18 7.9701 Tf 13.328 4.707 Td [(1 3 = 10 3 andthenalchargeofany 8 vertexisat least min f 8 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 3 ; 8 )]TJ/F15 11.9552 Tf 11.956 0 Td [(7 2 3 g = 10 3 Next,weconsiderverticesofdegree d with d 2f 10 ; 11 g .ByLemma5.11b, theseverticeshaveno 2 neighborswitha 7 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(neighbor.Thus,theseverticeshave nalchargeatleast d )]TJ/F18 7.9701 Tf 13.151 4.707 Td [(2 d 3 = d 3 10 3 ,since d 10 78
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Here,weconsiderverticesofdegree d with d 2f 12 ; 13 ; 14 g .Let v besuchavertex. ByLemma5.11b, v hasno 2 neighborwitha 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.ByLemma5.11b andLemma5.13, v hasatmosttwo 2 neighborseachhavinga 7 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(neighbor.Bythe dischargingrules, v hasnalchargeatleast d )]TJ/F15 11.9552 Tf 11.974 0 Td [(2 )]TJ/F15 11.9552 Tf 11.974 0 Td [( d )]TJ/F15 11.9552 Tf 11.974 0 Td [(2 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(2 3 = d )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 3 10 3 ,since d 12 Now,weconsiderverticesofdegree d with d 2f 15 ; 16 ; 17 g .Asbefore,Lemmas5.11band5.13guaranteethatsuchverticeshaveatmostone 2 neighborwith 7 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbors.Thus,theseverticesgiveatmost 1 4 3 + d )]TJ/F15 11.9552 Tf 11.072 0 Td [(1 2 3 charge.Hence,they havenalchargeatleast d )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 3 13 3 ,since d 15 Finally,consideran 18 + vertex v ofdegree d .Let r bethenumberof 1 neighbors of v .Let U = f u 1 ;u 2 ;:::;u q g bethesetof 2 neighborsof v .Foreach u i let N u i )]TJ 422.701 23.981 Td [(f v g = f u 0 i g .Let T = f u 0 i 2 U : d u 0 i 7 g andlet j T j = t .Since G [ T ] isplanar withgirthatleast5,Theorem5.10guaranteesatleast t +1 3 verticesin T thatforman independentset.ByLemma5.11b, d 18 r +11 t +1 3 +1 .Thus, d 18 r + 11 3 t + 14 3 : .1 Thenalchargeof v isatleast d )]TJ/F18 7.9701 Tf 13.049 4.707 Td [(7 3 r )]TJ/F18 7.9701 Tf 13.05 4.707 Td [(4 3 t )]TJ/F18 7.9701 Tf 13.05 4.707 Td [(2 3 d )]TJ/F20 11.9552 Tf 11.855 0 Td [(r )]TJ/F20 11.9552 Tf 11.854 0 Td [(t .Hence, v hasnalcharge atleast d 3 )]TJ/F18 7.9701 Tf 13.182 4.707 Td [(5 3 r )]TJ/F18 7.9701 Tf 13.182 4.707 Td [(2 3 t .From.1, d 3 )]TJ/F18 7.9701 Tf 13.181 4.707 Td [(5 3 r )]TJ/F18 7.9701 Tf 13.182 4.707 Td [(2 3 t 13 3 r + 5 9 t + 14 9 .When r 1 or t 4 thenalchargeisatleast 10 3 .When r =0 and t 3 ,thevertex v hasnalchargeat least d )]TJ/F18 7.9701 Tf 13.151 4.708 Td [(4 3 t )]TJ/F18 7.9701 Tf 13.151 4.708 Td [(2 3 d )]TJ/F20 11.9552 Tf 11.955 0 Td [(t d )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 3 12 3 ,since d 18 Theorem5.16 If G isaplanargraphwith girth G 6 ,then ` G 9 Proof: Let G beaplanargraphwithgirthatleast6andsuppose G isvertex minimalwith ` G > 9 .ByProposition5.3, mad G < 3 .Assigneachvertex v an initialchargeof d v andapplythefollowingdischargingrules: R1each 1 vertexreceives 2 chargesfromitsneighbor;and R2each 2 vertex 79
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awithone 8 + neighborandone 5 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighborreceives 1 chargefromits 8 + neighbor, bwithone 7 + neighborandone 4 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(neighborreceives 1 chargefromits 7 + neighbor, cwithone 6 + neighborandone 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighborreceives 1 chargefromits 6 + neighbor,or dreceives 1 = 2 chargefromeachneighbor,otherwise. Acontradictionwith mad G < 3 occursifthedischargingrulesreallocatechargeso thateveryvertexhasnalchargeatleast3;weshowthisisthecase. ByLemma5.11a,each 1 vertexhasa 9 + neighborandeach 2 vertexhas neighborswithdegreesumatleast 9 .Underthedischargingrules, 1 verticesand 2 verticesgaincharge 2 and 1 ,respectively,and 3 verticesneithergainnorlosecharge. Thus, 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(verticeshavenalcharge 3 ByLemma5.11b,each 4 vertex v hasno 1 neighborandhasatmostone 2 neighborwhoseotherneighborisa 6 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(vertex.Therefore,each 4 vertexhasnal chargeatleast 4 )]TJ/F18 7.9701 Tf 11.79 4.707 Td [(1 2 .Similarly,each 5 vertexhasno 1 neighborandhasatmostfour 2 neighborshavinganother 7 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(neighbor.Therefore,each 5 vertexhasnalcharge atleast 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 1 2 ,asdesired. If v isa 6 vertex,thenLemma5.11impliesthat v hasno 1 neighbor.Moreover, Lemma5.12impliesthatif v hassix2neighborsthenatmostoneofthemhasa 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.Hence, v haschargeatleast 6 )]TJ/F15 11.9552 Tf 12.553 0 Td [(max f 1+4 1 2 ; 6 1 2 g ,whichis3,as desired. SimilarlybyLemma5.11,a 7 vertex v hasno 1 neighbor.Moreover,byLemma 5.12,if v hasseven2neighbors,atmostoneofthemhasa 4 )]TJ/F16 11.9552 Tf 7.084 4.339 Td [(neighbor.Thus, v has chargeatleast 7 )]TJ/F15 11.9552 Tf 11.956 0 Td [(max f 2+4 1 2 ; 1+6 1 2 ; 7 1 2 g ,whichisatleast3asdesired. 80
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Finally,if v isa d vertexwith d 8 ,thenbyLemma5.11bwehave d 8 r +3 q +1 ; .2 where r isthenumberof 1 neighborsand q isthenumberof 2 neighborshavinga 5 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(neighbor.Thenalchargeon v isatleast d )]TJ/F15 11.9552 Tf 11.254 0 Td [(2 r )]TJ/F20 11.9552 Tf 11.254 0 Td [(q )]TJ/F18 7.9701 Tf 12.449 4.707 Td [(1 2 d )]TJ/F20 11.9552 Tf 11.254 0 Td [(r )]TJ/F20 11.9552 Tf 11.254 0 Td [(q = d 2 )]TJ/F18 7.9701 Tf 12.449 4.707 Td [(3 2 r )]TJ/F18 7.9701 Tf 12.449 4.707 Td [(1 2 q By.2, v hasnalchargeatleast 1 2 r +3 q +1 )]TJ/F18 7.9701 Tf 13.515 4.707 Td [(3 2 r )]TJ/F18 7.9701 Tf 13.515 4.707 Td [(1 2 q = 5 2 r + q + 1 2 .When r 1 or q 3 ,thisnalchargeisatleast 3 .Otherwise,when r =0 and q 2 v hasnalchargeatleast d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F18 7.9701 Tf 13.151 4.707 Td [(1 2 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2= d )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 2 3 ,since d 8 Theorem5.17 If G isaplanargraphwith girth G 7 ,then ` G 8 Proof: Let G beaplanargraphwithgirthatleast7andsuppose G isavertex minimalplanargraphwith ` G > 9 .ByProposition5.3, mad G < 14 = 5 .Assign eachvertex v aninitialchargeof d v andapplythefollowingdischargingrules: R1each1vertexreceives9/5chargefromitsneighbor;and R2each2vertex awithone 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighborandone 6 + neighborreceives4/5chargefromits 6 + neighbor, bwithone3neighborandone5neighborreceives1/5and3/5charge,respectively, cwithtwo4neighborsreceives2/5chargefromeachneighbor, dwithone4neighborandone 5 + neighborreceives1/5and3/5charge, respectively,or ewithtwo 5 + neighborsreceives2/5chargefromeachneighbor. Acontradictionwith mad G < 14 = 5 occursifthedischargingrulesreallocatecharge sothateveryvertexhasnalchargeatleast14/5;weshowthisisthecase. 81
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ByLemma5.11a,each1vertexhasan 8 + neighborandeach2vertexhas neighborswithdegreesumatleast8.Underthedischargingrules,1verticesand2 verticesgain9/5and4/5charge,respectively.If v isa3vertex,thenLemma5.11b impliesthat v hasatmostone2neighborwitha5neighborotherthan v .Thus, v givesatmost 1 = 5 charge.Hence, 3 )]TJ/F16 11.9552 Tf 7.084 4.339 Td [(verticeshavenalchargeatleast14/5. If v isa4vertex,thenLemma5.11bimpliesthat v hasatmostone2neighbor witha 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighborotherthan v .Thus, v hasnalchargeatleast 4 )]TJ/F15 11.9552 Tf 9.673 0 Td [(1 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(2 5 )]TJ/F15 11.9552 Tf 9.673 0 Td [(3 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(1 5 =3 If v isa5vertex,thenLemma5.11bimpliesthat v hasatmostone2neighbor witha 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.Thus, v hasnalchargeatleast 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(3 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(2 5 14 5 If v isa6vertex,thenLemma5.11bimpliesthat v hasatmostone2neighbor witha 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor,andatmostone2neighborhavinga 4 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(neighbor.Thus, v has nalchargeatleast 6 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(4 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(3 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(2 5 =3 If v isa7vertex,thenLemma5.11bimpliesthat v hasatmostone2neighbor witha 3 )]TJ/F16 11.9552 Tf 7.085 4.339 Td [(neighbor,andhasatmosttwo2neighborswitha 4 )]TJ/F16 11.9552 Tf 7.084 4.339 Td [(neighbor.Thus, v hasnalchargeatleast 7 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(4 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(3 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 )]TJ/F18 7.9701 Tf 6.675 4.977 Td [(2 5 = 17 5 If v isan8vertex,thenLemma5.11bimpliesthat v hasatmostone1 neighbor,atmosttwo2neighborswitha 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor,andatmosttwo2neighbors witha 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.Moreover,if v hasa1neighbor,then v doesnothavea2 neighborwitha 3 )]TJ/F16 11.9552 Tf 7.084 4.338 Td [(neighbor.Sincethedischargingrulesallocatechargetoneighbors withtheseconstraints, v hasnalchargeatleast 8 )]TJ/F15 11.9552 Tf 9.298 0 Td [(max 1 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(9 5 +7 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(2 5 ; 2 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(4 5 +6 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(2 5 = 17 5 If v isa d vertexwith d 9 ,thenLemma5.11bimpliesthat v hasatmost d 8 1neighbors,atmost d 4 neighborsthatareeithera1vertexora2vertexwitha 3 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor,andatmost d 3 neighborsthatareeithera1vertexora2vertexwith a 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(neighbor.Since v givesmorechargetoneighborsoflowdegree,weassume v hasasmanylowdegreeneighborsaspossible.Hence, v hasnalchargeatleast d )]TJ/F21 7.9701 Tf 11.186 4.707 Td [(d 8 )]TJ/F18 7.9701 Tf 6.675 4.976 Td [(9 5 )]TJ/F26 11.9552 Tf 9.991 9.683 Td [()]TJ/F21 7.9701 Tf 6.675 4.976 Td [(d 4 )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(d 8 )]TJ/F18 7.9701 Tf 14.147 4.976 Td [(4 5 )]TJ/F26 11.9552 Tf 9.991 9.683 Td [()]TJ/F21 7.9701 Tf 6.675 4.976 Td [(d 3 )]TJ/F21 7.9701 Tf 13.15 4.707 Td [(d 4 )]TJ/F18 7.9701 Tf 14.147 4.976 Td [(3 5 )]TJ/F26 11.9552 Tf 9.99 9.683 Td [()]TJ/F20 11.9552 Tf 5.48 9.683 Td [(d )]TJ/F21 7.9701 Tf 13.151 4.707 Td [(d 3 )]TJ/F18 7.9701 Tf 14.147 4.976 Td [(2 5 = 43 120 d ,whichisatleast3since d 82
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9 .Therefore,allverticeshavenalchargeatleast14/5andweobtainacontradiction. Wecalla d vertex lonely ifitisinexactlyonefaceof G .Wesaythatanonlonely 3 + vertex v is unique toaface f of G ifitisincidenttoacutedge uv suchthat d u > 1 and uv isalsoin f Lemma5.18 Let f beafaceinaplanargraph G with e c cutedgessuchthat f has s lonelyvertices,and t 3 + verticesuniqueto f .Wehave s + t 2 e c Proof: Weapplyinductionon e c .If e c =0 ,then s = t =0 andtheinequality holds.Inthefollowingtwocases,givensomeface f containingacutedge uv ,let G 0 bethegraphobtainedbycontractingtheedge uv toavertex w .Let f 0 betheface in G 0 correspondingto f .Let s 0 and t 0 bethenumberoflonelyverticesin f 0 andthe numberof 3 + verticesuniqueto f 0 ,respectively. Case1 : u or v islonely. Withoutlossofgenerality,assume u islonely.If v isalsolonely,then w islonely andtherefore s 0 = s )]TJ/F15 11.9552 Tf 12.112 0 Td [(1 .If v isnotlonely,then w isnotlonelyandstill s 0 = s )]TJ/F15 11.9552 Tf 12.112 0 Td [(1 Verticesuniqueto f arenotaectedbythecontraction,giving t 0 = t .Since f 0 has e c )]TJ/F15 11.9552 Tf 12.295 0 Td [(1 cutedges,theinductionhypothesisimpliesthat s 0 + t 0 2 e c )]TJ/F15 11.9552 Tf 12.294 0 Td [(1 .Therefore, s + t 2 e c Case2 : u and v areuniqueto f Since u and v arenotlonely, w isnotlonelyand s 0 = s .Aftercontracting uv ,either w isuniqueto f and t 0 = t )]TJ/F15 11.9552 Tf 12.149 0 Td [(1 or w isnotuniqueto f and t 0 = t )]TJ/F15 11.9552 Tf 12.149 0 Td [(2 ,whichyields t 0 +1 t t 0 +2 .Bytheinductionhypothesis, s 0 + t 0 2 e c )]TJ/F15 11.9552 Tf 11.771 0 Td [(1 .Since t t 0 +2 ,we have s + t 2 e c ,asdesired. Theorem5.19 If G isaplanargraphwith girth G 26 ,then ` G 3 Proof: Let G beplanarwithgirthatleast26andsuppose G isvertexminimalwith ` G > 3 .Assigneachvertex v aninitialcharge d v ,eachface f aninitialcharge 83
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l f ,andapplythefollowingdischargingrules: R1each 1 vertexreceives2chargesfromitsincidentfaceand1chargefromits neighbor; R2each 2 vertexreceives2chargesfromitsincidentfaceifitislonely,otherwise itreceives1fromeachincidentface; R3each 3 vertexwitha 1 neighborand aincidenttotwofacesreceives1chargefromeachincidentface,or bincidenttoonefacereceives2chargesfromitsface; R4each 3 vertexwithouta 1 neighborand aincidenttothreefacesreceives 1 3 chargefromeachincidentface, bincidenttotwofacesreceives 1 2 chargefromeachincidentface,or cincidenttoonefacereceives 1 chargefromitsface; R5each 4 vertexthathasa 1 neighborandis aincidenttothreefacesreceives 1 3 chargefromeachincidentface,or blonelyoruniquetosomeface f receives 1 chargefrom f ;and R6each 5 vertexthathastwo 1 neighborsandis aincidenttothreefacesreceives 1 3 chargefromeachincidentface,or blonelyoruniquetosomeface f receives 1 chargefrom f AcontradictionwithProposition5.9occursifthedischargingrulesreallocatecharge sothateveryvertexandfacehaschargeatleast4;weshowthisisthecase. Lemma5.11aimpliesthata1vertexhasa 3 + neighbor.Lemma5.11b impliesthata 4 )]TJ/F16 11.9552 Tf 7.085 4.338 Td [(vertexhasatmostone1neighbor,a5vertexhasatmosttwo 84
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1neighbors,andingenerala d vertexhasatmost d )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 2 neighborsofdegree1.Since verticesonlygivechargeto1neighbors, 6 + verticeshavenalchargeatleast4.Note thatif v isa d vertexwith d 2f 3 ; 4 ; 5 g ,atmost d )]TJ/F15 11.9552 Tf 12.452 0 Td [(3 neighborsofdegree1,and incidenttoatmosttwofaces,then v isuniquetoaface.Thus,allverticeshavenal chargeatleast4underthedischargingrules. Weturnourattentiontothenalchargeoffaces.ByTheorem5.5andthechoice of G G isconnectedandeachfacecontainsatleastonecycle.Therefore,eachface haslengthatleast26.Let R f bethesetofverticesincidenttoaface f thatareeither a2vertexora3vertexthatisnotlonelyandhasone1neighbor.Let f beaface with s lonelyvertices, t uniquevertices,and r verticesin R f .Lemma5.18implies that f hasatleast s + t 2 cutedges.Thus, l f 26+2 s + t: .3 AcombinationofthereduciblecongurationsinLemma5.14impliesthatthereare atmostfourconsecutiveverticesfrom R f inanycycleof f .Thus, r 4 5 l f )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t : .4 Bythedischargingrules, f hasnalchargeatleast l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.956 0 Td [(t )]TJ/F20 11.9552 Tf 11.956 0 Td [(r )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.956 0 Td [(t )]TJ/F20 11.9552 Tf 11.956 0 Td [(r = 2 3 l f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 3 s )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 r: By.4, 2 3 l f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 3 s )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 r 2 3 l f )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(4 3 s )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 t )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 4 5 l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t : .5 Therefore,thenalchargeof f isatleast 2 3 l f )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(4 3 s )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 t )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 4 5 l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t = 2 15 l f )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 s )]TJ/F20 11.9552 Tf 11.955 0 Td [(t ; whichisatleast4when l f )]TJ/F15 11.9552 Tf 12.336 0 Td [(2 s )]TJ/F15 11.9552 Tf 12.336 0 Td [(5 30 .When l f 2f 26 ;:::; 29 g ,.5gives nalchargeatleast4. 85
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Since[25]shows C n = C n for n 3 and G G ,wehave C 2 n +1 3 for n 13 .ByTheorem5.19,wehavethefollowingimmediatecorollary: Corollary5.20 If n 13 ,then ` C 2 n +1 =3 86
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