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Heuristic methods applied to difficult graph theory problems

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Title:
Heuristic methods applied to difficult graph theory problems
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Alalqam, Roqyah
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Denver, CO
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University of Colorado Denver
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English
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Graph theory ( lcsh )
Graph theory ( fast )
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non-fiction ( marcgt )

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The purpose of this study is to offer new insight and tools toward the pursuit of the largest chromatic number in the class of thickness t graphs. The thickness of a graph is the minimum number t of planar subgraphs where the graph can be decomposed. Determining the thickness of a given graph is known to be an NP - complete problem. The chromatic number is the smallest number k such that the vertices of a graph can be properly colored with k colors. Similarly, determining the chromatic number of a given graph is known to be an NP - complete problem. Another aim is to investigate the largest clique in the class of 3 dimensional visibility representation of graphs in which vertices are mapped to rectangles floating in R3 parallel to the x, y - axis, with edges represented by vertical lines or sight. In this thesis, heuristic search methods are combined with graph theoretic methods to discover more about the upper bound of the chromatic number for the Earth - Moon problem and its generalization to higher thickness and different surfaces. Additionally, they are used to explore explore about the upper bound of the largest size of the clique that has a rectangle visibility representation in 3 dimensions. By implementing heuristic search methods in Python and C languages on the sphere we found new 90 9 - chromatic thickness three graph, and 24 - chromatic thickness four graph. On the double-torus we found 14 - chromatic thickness two graph, 20 - chromatic thickness three graph, and 26 - chromatic thickness four graph, representation by rectangles in 3 dimensions we have found a representation of K22. Furthermore, various heuristic search techniques are c compared and detailed discussion of the heuristic search algorithms and their implementation is provided.
Thesis:
Thesis (M.S.)--University of Colorado Denver. Computer science
Bibliography:
Includes bibliographic references.
General Note:
Department of Computer Science and Engineering
Statement of Responsibility:
by Roqyah Alalqam

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Full Text
HEURISTIC METHODS APPLIED TO DIFFICULT GRAPH
THEORY PROBLEMS
by
ROQYAH ALALQAM
B.S., King Saud University, 2007
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
Master of Science in Computer Science
2012


This thesis for the Master of Science degree by
Roqyah Alalqam
has been approved for the Master of
Science in Computer Science by
Dr. Ellen Gethner, Advisor
Dr. Bogdan Chlebus
Dr. Michael Ferrara
November 5, 2012
n


Alalqam, Roqyah, R. (Master of Science in Computer Science)
Heuristic Methods Applied to Difficult Graph Theory Problems
Thesis directed by Associate Professor Ellen Gethner
ABSTRACT
The purpose of this study is to offer new insight and tools toward the pursuit of the
largest chromatic number in the class of thickness t graphs. The thickness of a graph is
the minimum number t of planar subgraphs where the graph can be decomposed.
Determining the thickness of a given graph is known to be an NP- complete problem.
The chromatic number is the smallest number k such that the vertices of a graph can be
properly colored with k colors. Similarly, determining the chromatic number of a given
graph is known to be an NP-complete problem. Another aim was to investigate the
largest clique in the class of 3-dimensional visibility representation of graphs in which
vertices are mapped to rectangles floating in R3 parallel to the x, y-axis. with edges
represented by vertical lines of sight.
In this thesis, heuristic search methods are combined with graph theoretic methods to
discover more about the upper bound of the chromatic number for the Earth- Moon
problem and its generalization to higher thickness and different surfaces. Additionally,
they are used to explore more about the upper bound of the largest size of the clique that
has a rectangle visibility representation in 3-dimensions. By implementing heuristic
search methods in Python and C languages on the sphere we found new 90 9-chromatic
thickness two graphs whose complements are Km- free. Also by applying the same
methods on the torus we found 19-chromatic thickness three graph, and 24-chromatic
thickness four graph. On the double- torus we found 1 Achromatic thickness two
graph, 20-chromatic thickness three graph, and 26-chromatic thickness four graph. On the
other hand, for the visibility representation by rectangles in 3-dimensions we have found
a representation of A22. Furthermore, various heuristic search techniques are compared
and detailed discussion of the heuristic search algorithms and their implementation is
provided.
The form and content of this abstract are approved. I recommend its publication.
Approved: Dr. Ellen Gethner
IV


DEDICATION
I dedicate this thesis to my wonderful family. Particularly to my parents, under-
standing and patient husband, Mahmoud, and to our gorgeous son Amjed, who is the joy
of our lives.
IV


ACKNOWLEDGEMENTS
I am deeply grateful to all those who willingly helped me accomplish this study. I
would like to express my sincere thanks and gratitude to my advisor, Professor Ellen
Gethner, for her guidance, support, motivation and encouragement throughout the period
this work was carried out. Her readiness for consultation at all times, her educative
comments, her advice during the writing of this thesis, and her concern and assistance
even with practical things have been invaluable.
I would like to express my gratitude to Dr. Thom Sulanke for his very useful comments
and suggestions. His sincere interest in this research theme is greatly appreciated. I
would like to thank the center for computational mathematics (CCM) for letting me used
the cluster computer to do my computations. I would also like to thank all the tutors of
the Department of Computer Science & Engineering at the University of Colorado at
Denver, who taught me during my study. The staff at the Department for their friendly
cooperation.
Vll


TABLE OF CONTENTS
Chapter
1. Introduction................................................................... 1
1.1 Statement of the Problems................................................ 3
1.2 Purpose of the Study.................................................... 4
1.3 Research Questions....................................................... 4
1.4 Significance of the Study................................................ 5
1.5 Limitations of the Study................................................. 5
2. Background of the Study...................................................... 6
2.1 Why Graphs?.............................................................. 6
2.2 Basic Structures and Definitions......................................... 6
2.2.1 Graphs........................................................... 7
2.2.2 Common Families of Graphs........................................ 8
2.2.3 Graph Operations ................................................ 9
2.2.4 Planar Graphs................................................... 10
2.2.5 r-Inflated Graphs .............................................. 12
2.3 Thickness of a Graph.................................................... 13
2.4 Graph Coloring.......................................................... 14
2.5 Visibility Representation............................................... 17
2.5.1 Bar-Visibility Graphs........................................... 17
2.5.2 Rectangle-Visibility Graphs..................................... 18
3. Review of Related Literature ...............................................19
3.1 Graph Coloring Problem.................................................. 19
3.1.1 Introduction.................................................... 19
3.1.2 Related Studies..................................................20
3.1.3 Conclusion...................................................... 23
vii


3.2 Visibility Representation Problem................................... 23
3.2.1 Introduction.................................................23
3.2.2 Related Studies..............................................24
3.2.3 Conclusion.................................................. 29
4. Optimization Search Methods...............................................30
4.1 Swarm Intelligence Algorithms (SI).................................. 30
4.1.1 Ant Colony Optimization Algorithm (ACO)..................... 31
4.1.2 Cuckoo Optimization Algorithm (COA) ........................35
4.1.3 Firefly Algorithm (FA)...................................... 38
4.2 Simulated Annealing Algorithm (SA)...................................41
5. Research Methodology .....................................................45
5.1 First Technique......................................................45
5.2 Second Technique ....................................................47
5.3 Third Technique .....................................................48
6. Findings and Discussions..................................................50
6.1 Results of the Study ................................................50
6.2 Heuristic Methods Comparison........................................ 67
6.3 Future Work......................................................... 70
Appendix
A. Experiment Results.................................................. 71
B. New Nine-Critical Graphs............................................ 74
Bibliography.............................................................. 121
viii


LIST OF FIGURES
Figure
1.1 Representing A6 by rectangles in 3-dimensions............................ 2
2.1 Simple Graph G........................................................... 7
2.2 Graph G and Subgraph H................................................... 8
2.3 Examples of Complete Graphs ............................................. 9
2.4 Complete Bipartite Graph K2>3 ............................ 9
2.5 A K4 Planar Graph....................................................... 10
2.6 Kuratowskis GraphK5 andA33 ............................. 12
2.7 Example of r-inflation Graphs........................................... 13
2.8 Decomposition of K9..................................................... 14
2.9 Four-colored map........................................................ 15
2.10 8-Chromatic Complete Graph A8 ............................. 16
2.11 1-Chromatic Empty Graph................................................ 16
2.12 2-Chromatic Bipartite Graph K3 4 ............................ 16
2.13 Bar-Visibility Graph .................................................. 17
2.14 Rectangles-Visibility Graph ........................................... 18
3.1 Orientable Surfaces of genus g=0, 1, and 2.......................... 19
3.2 Biembedding of A13 23
3.3 A8 has a rectangle-visibility graph ..............................25
3.4 Representing Kg by K4 VK5 blocks ..............................26
3.5 Representation ofK2o ....................................................26
3.6 Representing A7 by unit squares ..............................27
3.7 Any Km has a rectangle-visibility representation ......................27
3.8 Any Kn has a discs-visibility representation ............................28
4.1 Ants find shortest path between their nest and food sources............. 32
4.2 Pseudo code of ACO algorithm ............................. 33
IX


4.3 Example for Applying ACO Algorithm ................................33
4.4 Trace for Applying ACO Algorithm ..................................35
4.5 Pseudo code of CS algorithm .......................................36
4.6 Example for Applying CO A Algorithm .................................37
4.7 Trace for Applying CO A Algorithm ...................................38
4.8 Pseudo code of FA algorithm .......................................39
4.9 Example for Applying FA Algorithms ..................................40
4.10 Trace for Applying FA Algorithms ....................................41
4.11 Pseudo code of SA algorithm .......................................42
4.12 Example for Applying SA Algorithms ................................43
4.13 Trace for Applying SA Algorithms ..................................44
6.1 New 9-Critical Graph whose Complement is AT3-free ...................52
6.2 New 9-Critical Graph whose complement is AT4-free ...................53
6.3 New 9-Critical Graph whose complement is AT5-free ...................53
6.4 Decomposition of A19 56
6.5 Decomposition of A25 57
6.6 Decomposition of A24 58
6.7 Decomposition of (4,4,4,4,3)-inflated C5 ...........................65
6.8 Decomposition of 4-inflated C7 .................................... 66
6.9 First Technique Algorithms Performance (1) 67
6.10 First Technique Algorithms Performance (2) 68
6.11 First Technique Algorithms Performance (3) 68
6.12 Third Technique Algorithms Performance ............................69
B. 1.1 Ant Colony Algorithm (ACO).......................................74
B. 1.2 Firefly Algorithm (FA)........................................... 79
B.1.3 Simulated Annealing Algorithm (SA)................................ 84
B.2.1 Ant Colony Algorithm (ACO)........................................90
B.2.2 Firefly Algorithm (FA)............................................ 95
XI


B.2.3 Simulated Annealing Algorithm (SA)............................. 100
B.3.1 Ant Colony Algorithm (ACO)..................................... 105
B.3.2 Fir efly Algorithm (FA)........................................110
B.3.3 Simulated Annealing Algorithm (SA)............................. 115
XI


LIST OF TABLES
Table
1 The functions H(g) for a surface of genus g ................................21
2 The functions B(g) for a surface of genus g ............................... 22
3 Lower and upper bounds for rectangle-visibility representations .... 28
xii


1. Introduction
A graph is an abstract representation of relationships. It is defined as a diagram
consisting of points, called vertices, some of which are connected by lines, called edges.
For example, assigning registers to temporary variables can be converted into an
equivalent graph by letting each variable to be a vertex and connecting two vertices by an
edge if the corresponding variables are interfere. Suppose we are asked to help out a
compiler to reduce the number of registers needed in such a way that if there is an edge
between two variables, then we should not assign the same register to them since this
register needs to hold the values of both variables at one point of time. The idea is to use
the fewest possible number of registers. Can this be done?
Here, we briefly introduce the graph coloring problem, a classic problem in graph theory.
The graph coloring problem is to assign colors to the vertices of a graph in such a way
that no two vertices connected by an edge share the same color, and the aim is to use the
fewest possible number of colors.
Another example, the circuit board can be converted into an equivalent graph by letting
the components to be vertices and the wires that connected certain pairs of the components
to be edges. Suppose we are asked to help out a circuit designer to model the circuit
board in such a way that connecting wires lie vertically or horizontally and do not
cross and that the components lie perpendicular to the wires. Can this be done?
The type of graph that is most likely to help the circuit designer is a rectangle-
visibility graph. From this point, we introduce the visibility representation problem for
graph. The visibility representation problem is to determine whether a graph is
representable or not.
The graph coloring problem has an essential role in computer science. It models major
real-world problems. The major application areas are: time tabling and scheduling,
frequency assignment, register allocation, and printed circuit testing. On the other hand,
the visibility representation problem has also large number of applications related to
computer science such as VLSI design, CASE tools, circuit board layout, and animation
problem.
In this thesis, we study a generalization of a problem posed by Gerhard Ringel in
1


1959 [Rin59], which is a natural generalization of the Four-Color problem. It is a long
standing open problem, which is determining the largest chromatic number of an
arbitrary thickness t graph for any t > 2 and any genus g. In addition, we study 3-
dimensions visibility that represent vertices by 2-dimensional rectangles placed in
planes parallel to the xy-plane. Two vertices are connected by an edge if and only if
they can see each other in the direction that is perpendicular to the z-axis. This type
of representation was introduced as a generalization of the 2-dimensional visibility
representation [BEF+94], The 3-dimensional rectangle visibility representation received
a wide attention. However, we focus on the maximum size of a complete graph with a 3-
dimensions visibility representation by rectangles. See figure 1.1 for an example for
visibility representation of complete graph K6 by rectangles in 3-dimensions.
Figure 1.1: Representing K6 by rectangles in 3-dimensions
This thesis presents the results of using heuristic search methods implemented in Python
and C languages to find ^-chromatic thickness 1 graphs on n vertices and genus g of
orientable surface, and to find the representation of largest complete graph Kn by
rectangles in 3-dimensions. Additionally, a variety of different heuristic search methods
are compared and detailed discussion of the heuristic search algorithms is provided.
The structure of this thesis is going to be as follows: Chapter 2 presents the relevant
background information in the following order: why graphs?, basic structures and
definitions, thickness of graphs, graph coloring, and visibility representation for graphs.
Chapter 3 is a review literature of related studies.
2


In Chapter 4, a presentation of the different heuristic methods will be presented. In
Chapter 5, we are finally moving into the main part of this thesis, namely the
methodology of the study, implementation and a description of the code used to
accomplish this study. The closing chapter, Chapter 6, will contain the results of this
study and a discussion of the results. Additionally, a comparison of the different
heuristic methods for two different problems and suggestions for future work will be
provided. There will be an Appendix A that summarizes the results of the experiment,
and Appendix B contains 90 9-critical graphs with thickness-two and genus 0.
1.1 Statement of the Problems
Let 0(G) to be the thickness of graph G. A graph G has thickness t with respect to the
genus g, if G can be decomposed into t subgraphs and no fewer than t. This partition makes
t copies of the vertices of G and each edge of G is assigned to one of the t copies.
Moreover, each subgraph of G should be g embeddable. Let G/,g, denotes an orientable
graph with thickness-/ and genus g. Let G be a graph; the r-inflation of G is the
lexicographic product G[Kr ], and is denoted by G[r], The chromatic number of a
graph G, denoted/(G), is the minimum number of colors needed to color the vertices of
G such that no two adjacent vertices receive the same color. The question we want to
study is:
What is the best upper bound for the chromatic number of any genus
g thickness t orientable graph?
Visibility representations of graphs map vertices to sets in Euclidean space and express
edges as visibility relations between these sets. For graph G = (V, E) the 3-dimensional
visibility representation by rectangles is defined as an arrangement of disjoint rectangles
in R3 such that the planes determined by the rectangles are vertical to the r axis, and
the sides of the rectangles are parallel to the x axes or y axes. A given graph G is said
to be representable if and only if its n vertices can be associated with n disjoint
rectangles parallel to the x and y axes in R3 such that vertex v, and v, are adjacent in
G if and only if their corresponding rectangles R, and R, are r visible. The question
we want to study is:
What is the upper bound on the size of the largest clique that
can be represented in 3-dimensions?
The heuristic search methods are applied to find the upper bound of the chromatic
number of any genus g thickness t orientable graph. In addition, they are applied to
find the maximum size of a complete graph with 3-dimensions visibility representation
by rectangles.
3


1.2 Purpose of the Study
There ate many interesting, unsolved coloring problems for graphs that can be decomposed
into multiple planar layers. The decision problem thickness, which determines for an
input graph G and integer t if 0(G) > I, is NP-Hard[Mit81], Furthermore, the decision
problem ^-colorable graph, which determines if there is a mapping of k colors to
vertices such that all adjacent vertices have different colors, is NP-Hard [GJ79], The
combination of these two NP-Hard problems makes solving multi-thickness graph
coloring problems very challenging. Thus, it would be desirable to have a collection of
graph families for which both the thickness and chromatic number are understood such
as complete graphs Kn. In this thesis, heuristic search methods are combined with graph
theoretic methods to generate graphs with specific chromatic numbers or thickness. The
goal of creating and exploring these graphs is to discover more about the bounds of
the chromatic number for the Earth-Moon problem and its generalization to higher
thickness and different surfaces.
Furthermore, several data presentation problems involve drawing graphs. The study of
this drawing was originally motivated by VLSI layout. A significant problem is the
one of determining visibilities between different electrical components. Since VLSI
structures are laid in a plane, and the components are bounded by isothetic oriented
rectangles, then the visibilities can be studied within the two families of parallel sides,
independently [LMW87], The purpose of the study is to represent the problem of
segment visibility in graph form and to discover more about the bounds of the largest
size of complete graphs that have rectangles visibility representation in 3-dimensions by
combining heuristic search methods with graph theoretic methods.
1.3 Research Questions
The major questions that are addressed by this study are the following:
1. Does there exist a graph G2>0 with /(G2j0) = 10,11, or 12?
2. Does there exist a graph G3>0 with /(G3 0) = 17 or 18?
3. Does there exist a graph G4>0 with /(G4j0) = 23 or 24?
4. Does there exist a graph G3 i with /(G3 ,) = 19?
5. Does there exist a graph G4>i with /(G4 ,) = 25?
6. Does there exist a graph G2>2 with /(G2>2) = 14?
4


7. Does there exist a graph G3>2 with /(G3>2) = 20?
8. Does there exist a graph G4 2 with /(G4>2) = 26?
9. Does the 10-chromatic /4,4.4,4.3/-inflated C5 have thickness two?
10. Dose the 10-chromatic 4-inflated C7 have thickness two?
11. Is there a rectangle-visibility representation for K23 in 3-dimensions?
1.4 Significance of the Study
Except for the toroidal graphs whose surface has Euler characteristic 0 [BM92], the
Earth-Moon problem and its generalization to higher thickness and different surfaces it
has not been solved yet. So, we attempted to reach new results and solve the problem
by this study.
Furthermore, the current best upper bound on the size of the largest complete graph Kn
that can be represented by rectangles is 50 [BDHS97][S09], and the lower bound is 22
[BEL+93], Throughout this study we attempted to narrow the gap between the
known upper and lower bound for representation of Kn by rectangles.
1.5 Limitations of the Study
Finding the largest chromatic number of any genus g thickness-t orientable graph is
complicated for the reason that for given an arbitrary graph G and a fixed positive
integer t > 2, and genus g > 0, verifying that 0(G) = t is an NP-complete problem.
Similarly, given an arbitrary graph G and fixed positive integer k > 2, verifying that
/(G) = k is also NP-complete. In that case, the approach of starting with a graph of
known thickness and finding its chromatic number, or vice versa, will not often end in
achievement [ABG10],
Since bar visibility graphs are naturally planar [Wis85] [TT86], and rectangle
visibility graph is the union of two bar visibility graphs, then rectangle visibility
graph naturally has thickness at most two, seen by partitioning the edges into two sets
corresponding to vertical and horizontal visibilities. It would be useful to have a simple
characterization of rectangle visibility graphs, but no characterization has yet been found;
neither has the problem been shown to be NP-complete, though it is an NP-complete
problem to recognize thickness-two graphs [DH97],
5


2. Background of the Study
This chapter gives a brief introduction to the terminology used later in this thesis. The
aim is to provide the reader with the minimal necessary graph theoretic background to
understand the concepts and results of this dissertation.
2.1 Why Graphs?
Fundamentally, computer science is a science of abstraction as Aho and Hopcroft
illustrated [AH74], Therefore, computer scientists must create abstractions of real-
world problems that can be represented and manipulated in a computer. For example, for
successful scheduling of final exams, we have to consider the associations between courses,
students and rooms. Such set of connections between items is modeled by graphs. A graph
is composed by some elements called vertices, and the relations among them, are called
edges.
The basic idea of graphs was introduced in 1736 by the great mathematician Leonhard
Euler. He used graphs to solve the famous Konigsberg Bridge problem. Euler studied the
problem of Konigsberg Bridge and constructed a structure to solve the problem which
is called Eulerian graph [SHL07], In 1840, A.F Mobius gave the idea of Complete
Graph and Bipartite Graph. Euler and Kuratowski formulated their famous
characterization of Planar Graphs. Euler mentioned his formula in a letter to Goldbach
in 1750, and then proved it for convex polyhe- dra in 1752. Kuratowskis
observations captured all non-planar graphs, and in 1930 he published a proof of his
well-known graph planarity criterion [WesOl], In 1852, Gutherie found the famous
Four-Color problem [Mit81], Even though the Four-Color problem was invented in
1852, it was solved only after a century by Kenneth Appel and Wolfgang Haken
[AH77], Furthermore, the Four-Color prob- lem has been changed to Four-Color
theorem; the last word on the Four-Color problem has not been said anymore. This time
is considered as the birth of Graph Theory. [Deo74],
2.2 Basic Structures and Definitions
In order to simplify the reading, we have to state the basic standard notation that we
will use throughout the work. We will use the notation given in textbook
Introduction to Graph Theory by Douglas B. West [WesOl] and Discrete
Mathematics with Algorithms by M. Albertson and J. Hutchinson [AH88],
6


2.2.1 Graphs
Definition 1 [WesOl] A graph G = (V, E) is a combinatorial object composed by a
pair, where V is the set of vertices andE is the set of edges. We consider a graph to be
simple if E is not a multiset. Usually, the set of vertices is labeled to represent graphs
with points as vertices and lines linking these points as edges.
See figure 2.1 [WesOl] for simple graph G with V = {\, 2, 3, 4, 5, and 6} and E
= {(1,2), (1,5), (2,3), (2,5), (3,4), (4,5), (4,6))
Figure 2.1: Simple Graph G
n = |V and e = |E| denote the number of vertices and edges respectively. Note
that:
< The set of edges that contains v is E(v). The vertices that share an edge will be called
neighbors, and the set of neighbors of v denoted by N(v).
Definition 2 [WesOl] The degree of a vertex v is the number of vertices adjacent to v (or
equivalently, the number of edges incident with v). We denote the degree of v by deg(v,
G) or deg(v). The degree sequence of a graph is the sequence formed by arranging the
vertex degrees in decreasing order.
In figure 2.1, the degree of the vertices is as follows: deg(l)= 2, deg(2)= 3, deg(3)= 2,
deg(4)= 2, deg(5)= 3, and deg(6)= 1. And the degree sequence ={3, 3, 2, 2, 2, 1)
Theorem 1 (Handshaking Theorem) [WesOl]
IfV(G) = (vh v2, ..., v}, then
E"=i deg(vi) = deg(v\) + ... +deg(vn) = 2 \ E \
7


Proof The sum of the degrees counts the total number of times an edge is incident with
a vertex. Since the degree of a vertex is the number of edges incident with that vertex.
Also, every edge is incident with two vertices, each edge gets counted twice, one at each
end. Thus the sum of the degrees equals twice the number of edges.
Definition 3 JAH88L4Jiraph H is a subgraph of a graph G, denoted by H t^G, if V(H)
^V(G) andE(H) <=E(G).
Figure 2.2 [AH88] shows H subgraphs of graph G.
Figure 2.2: Graph G and Subgraph H
Definition 4 [AH88] An independent set of vertices in a graph is a set of mutually
non-adjacent vertices. The independence number of a graph G is the maximum
cardinality of an independent set of vertices. It is denoted by a(G).
2.2.2 Common Families of Graphs
Definition 5 A graph in which every pair of distinct vertices is joined by an edge is called
complete graph. A complete graph with n vertices is called an n-clique and is denoted by
Kn-
Figure 2.3 shows examples of complete graphs.
Theorem 2 An n-clique graph has exactly n (n-l)/2 edges.
Proof In Kn each vertex has degree n 1. Thus the sum of the degrees equals n(n 1).
By (Handshaking Theorem), this sum also equals 2\E\. Thus 2\E\ = n(n 1) and A =
n(n \)/ 2.
8


Definition 6 A graph G is called complete bipartite graph, if V(G) has a partition
to two subsets X andY such that each edge lu, vj £ G connects a vertex of X and a vertex
of Y In this case, (X,Y) is a bipartition of G, andG is (X, Yj- bipartite.
Figure 2.4 [AH88] shows complete bipartite graph K2,3 which has two bipartition
subsets X = 2 and 7 = 3.
Figure 2.4: Complete Bipartite Graph K2,3
Definition 7 A cycle graph is a graph that consists of a single cycle, or in other words,
some number of vertices connected in a closed chain, denoted Cn where n is the number of
vertices. The number of vertices in Cn equals the number of edges, and every vertex has
degree 2; that is, every vertex has exactly two edges incident with it.
2.2.3 Graph Operations
The union of two graphs is formed by taking the union of the vertices and edges of the
graphs.
The join G V H of the graph G and H is obtained from the graph union G u H and
adding an edge between each vertex of G and each vertex of H.
9


The product G = G\ x G2has V(Gi) x V(G2), and two vertices (nh u2) and (vb v2)
of G are adjacent if and only if either u] = u2 and u2v2 EE{G2) or u2 = v2 and uxvx E
E(Gi).
Vertex Removal: If v, is a vertex of a graph G = (V, E), then G v, is the induced
subgraph of G on the vertex set V v,; that is, G v, is the graph obtained after removing
from G the vertex v, and all the edges incident on v,.
Edge Removal: If e, is an edge of a graph G = (V, E), then G e, is the subgraph
of G that results after removing from G the edge e,. Note that the end vertices of e, are
not removed from G.
2.2.4 Planar Graphs
Planar graphs are those that can be drawn in the plane so no two edges cross, except
possibly at the endpoint of the edges. Graphs arising in many applications are planar by
definition, such as maps of countries. Others are planar by accident, like trees. [Ski08]
Definition 8 [AH88] A graph G is planar if there exists a drawing of G in the plane
in which no two edges intersect in a point other than a vertex of G.
Recall what K4 looks like (Square with edges crossing in the center so that all vertices
are adjacent), see figure 2.3. K4 has edges which intersect at non-vertex locations.
Therefore in its original state K4 is not planar, but K4 is isomorphic to the graph below
[AH88], which is planar. Therefore K4 is planar.
A simple, connected, planar graph splits the orientable surface into a number of
regions, including totally enclosed regions and one infinite external region. Euler
Figure 2.5: A K4 Planar Graph
10


observed a relationship between the number v of vertices, the number e of edges, and the
number / of regions (faces) in such a graph [Ger98], This relationship is known as
Eulers formula.
Theorem 3 (Eulers Formula) If a finite, connected, planar graph is drawn on genus
g of orientable surfaces without any edge intersections, and v is the number of vertices, e
is the number of edges and f is the number of faces, then
v-e+f = 2-2g
Proof by induction: If G is acyclic, then f= 1, and the theorem holds because then G
is a tree and e = v 1. Otherwise G has a cycle. Let x be an edge in a cycle. Deleting
x from G and its planar drawing results in a graph G with v vertices, e 1 edges and
f~ 1 faces (since deleting an edge involved in a cycle merges the two faces on either side
of it). By induction, we have v (e 1) + {f 1) = 2 2g, and so v e +f=2 2g.
Eulers formula by itself does not provide us with a tool for showing that some graphs
do not embed on a surface of genus g, because it refers to the set of faces / 'in a prospective
planar drawing of the graph. But we can use it to derive the following sufficient principle
for non-planarity. Let e(g) denote the Euler characteristic of genus g. It is well-known
that e:(g) = 2-2g for orientable surfaces.
Corollary 1 Suppose G is a connected planar graph, with v nodes, e edges, and./ faces,
where v >3. Then e <3(v e).
Proof The sum of the degrees of the faces is equal to twice the number of edges from
(Handshaking Theorem). But each face must have degree >3. So we have ?/ -2c.
Eulers formula says that v-e+J~= 2-2g, sof=e~v+e and thus 3 f= 3e-3v+e.
Combining this with 3yf <2e, we get 3e 3v + s <2e. So e <3(v e).
Lemma 1 In any orientable graph G with Euler chamcteristic e and thickness-t, there
is a vertex of degree at most (6 )t\
Proof From Corollary 1, the maximum number of edges in each planar layer is at most
e <3(v-e).
If G is an orientable graph with Euler characteristic e and V vertices has thickness- /, then
each layer has at most 3(v s) edges. In total, G has at most 3(v e)t edges. Thus, e <
3(y-e)t. From handshaking theorem (theorem 1), we can find the average degree:
11


Hfc=i deg(vi) = 2|e|. thus:
dcg(Vj)
2^1=1 r
Hence, at least one vertex has a degree of |(6 )t\ or less.

In addition, we can tell whether a given graph is planar or not by a helpful fact that the
two graphs given in figure 2.6 are both non-planar. These are the Kuratowiski graphs.
Thus, any non-planar graph must contain a subgraph closely related to one of these two
graphs.
Theorem 4 (Kuratowskis Theorem) a graph is planar if and only if it does not
contain any subdivision of Ks or 3 as a subgraph.
See [WesOl] for the proof of Kuratowskis Theorem.
2.2.5 r-Inflated Graphs
Definition 9 [ABG10] [ABG11] Let G be a graph; the r-inflation of G is the
lexicographic product G[Kr \ andis denoted by G[r]. G[2] is called the clone of G.
See figure 2.7 for an example of r-inflated graph.
By definition, we obtain G[r] by replacing each vertex of G by Kr and each edge of G
by K2r (which contains a Kr for each vertex of the edge). An r-inflation of G has the
following properties [ABG10] [ABG11]:
(a)
(b) K% 3
Figure 2.6: Kuratowskis Graphs Ks and AT33
12


(a) If the number of vertices and edges of G are V and /? respectively, then the
y
number of vertices and edges of G[r] are rV and V + r2E respectively.
(b) G[.v/-] = (G[.v])[r], For any complete graph Ks and any positive integer r, we have
Ks[r] = Ksr.
(c) Independence is invariant under inflation. That is, if the independence number of G
is a then the independence number of G[r] is a as well.
(d) If the clique number of G is a>, then the clique number of G[r] is rco.
(e) If the chromatic number of G is / then the chromatic number of G[r] is at
most ry.
(f) Any edge of G induces a Klr'va G[r],
2.3 Thickness of a Graph
By definition, if a graph is planar then it can be embedded in a single plane. Assume
that we are given a non-planar graph. How many planes are necessary in order to fully
embed it? This idea leads to the next definition.
Definition 10 The thickness of a graph G, denoted by Q(G) is the minimum number
of planar subgraphs whose union is G and is a measure of its degree of non-planarity.
Determining the thickness of a graph is NP-complete problem [Mit81], The graph classes
with well-known thicknesses are the complete graphs, complete bipartite graphs, and
hypercubes. The thickness of complete graphs Kn is solved for al- most all values of n
by Beineke and Harary. A decade after that Alekseev and Gonchakov, solved the
remaining cases [AMS96],
13


Theorem 5 [Ski08] For complete graphs, Q(K) = ^ with the exception that
{K9) = {KW) = 3.
See figure 2.8 [AMS96] for a decomposition of K<> into three planar subgraphs.
Figure 2.8: Decomposition ofW9
2.4 Graph Coloring
Graph coloring is one of the classic problems in graph theory. It has several appli- cations
such as scheduling, registering allocation in compilers, assigning frequency in Mobile
radios, etc. A graph may be colored in several ways. We may color either the vertices, the
edges, or the faces. We will focus on coloring the vertices. The rule is to assign a color
to each vertex in such a way that two connected vertices do not have the same color.
Therefore, the graph is properly colored graph and the minimum number of colors used to
color a graph is called the chromatic number. Among problems dealing with coloring,
the most famous one is called Four-Color problem which asks if it is possible to color any
planar graph with at most four col- ors. This problem was set out by Francis Guthrie in
1852 [Mit81], he noticed that four colors are sufficient to color the map of counties of
England. In 1976, Kenneth Appel and Wolfgang Haken published a first proof [AH77],
Their proof was based in ruling out a lot of configurations using a computer, because it
was the only way to do it in a reasonable time. Despite of this, it is assumed that the proof
is correct.
14


A graph can be constructed from any map, the regions being represented by the vertices
of the graph and two vertices being joined by an edge if the regions corresponding to the
vertices are adjacent. The resulting graph is planar. Therefore, a map is corresponds to a
planar graph. The Four-Color Theorem emphasizes that any map can have its faces colored
with at most four colors such that no two faces that share a piece of boundary have the
same color. See figure 2.9 for a 4-colored map of the United States.
Figure 2.9: Four-colored map
Definition 11 A graph G is k-colorable if we can assign the colors {1, 2, 3, ..., k} to the
vertices in V(G), in such a way that every vertex gets exactly one color and no edge in
E(G) has both of its endpoints colored with the same color. Ifk is the smallest number
such that G admits a k-coloring, we say that the chromatic number of G is k, and write
X{G) =k.
Example 1 Illustrate how the above definition works:
(a) The complete graph K with n vertices has chromatic number//. To see that take
any graph of Kn, and look at any vertex v, its connected to every other vertex, it
cannot be the same color as any other vertex. Therefore must have a different color
than every other vertex, which forces n colors. See figure 2.10 for an example for
8-chromatic complete graph K^.
(b) The empty graphs, which a graph with no edges, its chromatic number is 1.
Since we can color all vertices with the same color. The empty graphs are the only
graphs with chromatic number 1. Simply, any graph with an edge needs at least two
colors to properly color it, as both endpoints of that edge cannot be the same color.
See figure 2.11 for an example for 1-chromatic empty graph with five vertices.
15


Figure 2.11: 1-Chromatic Empty Graph
(c) The bipartite graphs by definition, every bipartite graph with at least one edge has
chromatic number 2. Since these graphs are contain two sets. See figure 2.12 for an
example for 2-chromatic bipartite graph /Cu.
16


Theorem 8 Let G be any graph with maximum vertex degree d. Then G is properly (d +
l)-colorable.
Proof Assume that all graphs on k or fewer vertices with maximum degree d are (d +
l)-colorable. Consider G, a graph with maximum degree d, on k + 1 vertices. Find
vertex V of degree d in G, and delete it; call the new graph G By induction, we can
properly color G with (d + 1) or fewer colors. Then replace V and its edges to get G
back. Color V with any color from among the (d + 1) we started with that has not been
used on the neighbors of V. Thus, by induction G is (d+ l)-colorable.
Definition 12 A graph is k-critical if it is k-chmmatic but every proper subgraph can be
properly colored with fewer than k colors.
2.5 Visibility Representation for Graph
The idea of representing a graph using a visibility representation (VR) was introduced
in the 1980s as a model tool for VLSI wire routing and circuit board layout [TT86],
In the VR for a graph, the vertices map to objects in Euclidean space and the edges are
determined by certain visibility relations.
2.5.1 Bar-Visibility Graphs
Bar-visibility graph for a graph G is represented by horizontal line segments, such that
the vertices of G are represented by non-overlapping horizontal segments called vertex
segments, and if the corresponding segments (adjacent vertices) are vertically visible, we
can say that the graph is a bar representable [DEG+05], Figure 2.13 shows an example of a
bar-visibility representation with the graph it induces. This form was introduced by
Luccio, Mazzone and Wong in 1983 [LMW87],
Figure 2.13: Bar-Visibility Graph
17


2.5.2 Rectangle-Visibility Graphs
The rectangle-visibility graph for a graph G is represented by drawing a graph in the
plane so that the vertices of the graph are the rectangles and the edges are horizontal or
vertical line segments. We say that the graph is a rectangles- visibility graph if it has a
visibility drawing. Figure 2.14 [Bei97] shows an example of a rectangle-visibility
representation with the graph it induces [BDHS97],

R,
Figure 2.14: Rectangles-Visibility Graph
18


3. Review of Related Literature
In this chapter, we review some existing theories and problems on graph coloring, and
visibility representation for graphs. The chapter is divided into two sections. In section
3.1, we briefly state the currently famous theoretical results regarding the graph
coloring problem. In section 3.2, we state the well-known theoretical results concerning
the visibility representation problem for graphs in 2-dimensions and 3-dimensions. This
is done to give a framework in which can be considered for those two problems.
3.1 Graph Coloring Problem
3.1.1 Introduction
Let 0(G) to be the thickness of graph G. A graph G has thickness t with respect to the
surface S, if G can be decomposed into t subgraphs and no fewer than t such that each
of which can be embedded on S. This partition makes t copies of the vertices of G and
each edge of G is assign to one of the t copies. Moreover, each subgraph of G should be
embeddable on that surface. The chromatic number of a graph G, denoted /(G), is the
minimum number of colors needed to color the vertices of G such that no two adjacent
vertices receive the same color. Let e(g) denote the Euler characteristic for genus g of
orientable surfaces. The sphere is genus g=0, the torus is genus g=l, and the double-
torus is genus g=2. Let Gtjg, denote an orientable graph with thickness t and genus g.
It is well-known that
e:(g) = 2-2 g for orientable surfaces [JT95], See figure 3.1 for genus of orientable surfaces.
Figure 3.1: Orientable Surfaces of genus g=0, 1, and 2
In 1959, Ringel asked: What is the chromatic number of G2,0, any thickness two and
genus 0 orientable graph? [Rin59], The question in general:
19


What is the best upper bound of the chromatic number of Gtg any
thickness t and any genus g orientable graph?
3.1.2 Related Studies
The Four-Color theorem has been investigated in many research papers and motivated
numerous coloring problems study which are still alive and the source of challenging
problems and theories, as illustrated by Jensen and Toft [JT95], All these problems have
a basic question: how to color a graph in such a way that two adjacent vertices do not
share a color?
For a century and a half, the Four-Color problem has been played the foremost role in
the development of graph theory. It dates back to 1852, when Thomas Gutherie was
trying to color the map of counties of England; he noticed that four colors suffice. Fie
asked his brother Frederick Gutherie if any map can be colored using only four colors, so
no adjacent regions colored with the same color. Then, Frederick explained the problem
to his teacher August DeMorgan, who in turn showed it to Arthur Cayley. The
problem at the beginning published as a puzzle by Cayley in 1878. Later in 1879, A.
B. Kempe gave the first published proof of the Four-Color Conjecture. In 1890,
Heawood pointed out a serious flaw in Kempes proof. A century later, Appel and
Haken solved the Four-Color problem [Mit81] [AH77],
At the time in 1970s, the proof by Appel and Haken has not been fully accepted. Their
proof was based in ruling out different types of graphs, (configurations) using a
computer, because it was the only way to do it in a sensible time. In spite of this, it
was assumed that the proof is correct [Mit81],
Recently, four mathematicians at Ohio State University and Georgia Institute of
Technology (Robertson, Sanders, Seymour and Thomas), gave a new proof of the Four-
Color Theorem for a planar graphs. Their proof idea is similar to Appel and Hakens
proof. They found a set of 633 configurations, and proved that none of them can appear
in a smaller counterexample. Thus, they proved that no counterexample exists
[RSST96],
Actually, coloring graphs on surfaces has been studied extensively. Heawood [Hea90] showed
that each graph embedded in such a surface of genus g > 1 has chromatic number at most:
H(g) = L^I
J
20


This upper bound is known as theHeawood number H(g). Table 1 shows the function
of H(g) for a genus g [SS06],
Table 1: The functions H(g) for a surface of genus g
g 1 2 3 4 5 6 7 8 9 10 11 12
H(g) 6 7 7 8 9 9 10 10 10 11 11 12
Later, in 1968 Ringel and Youngs [RY68] found the corresponding lower bounds, by
proving that the complete graph on H(g) vertices can be embedded on any surface of
Eulerian genus g, with the exception of the Klein bottle, where the correct bound on the
chromatic number is 6 (Heawoods formula gives 7) as verified by Franklin [Fra34],
The Empire problem also known as the (M-pire problem) asks for the maximum
number of colors needed to color countries such that no two countries sharing a
common border have the same color where each country consists of M disjoint regions.
Heawood [Hea90][JR85] showed in 1890 that theM-pire chromatic number for any genus
with Euler characteristic s, with the exception of (M = 1 and s = 2, Four-Color
theorem), at most:
Heawood [Hea90] showed that 6M colors are sufficient, and for the case M = 2 the
bound was sharp. Furthermore, in 1981 H. Taylor found that the Heawoods bound was
sharp for M =3 [Gar05], Jackson and Ringel showed that 6M colors are sufficient for
all M > 1 [JR85],
The Earth-Moon problem was so coined by Ringel, which is a special case of the M -pire
problem for countries with M = 2 disjoint regions, with one region of each country lying
on the Earth and one on the Moon. In particular, it is the search for the largest
chromatic number of any graph G2jo (thickness two and genus 0). Ringel [Rin59]
remarked that the few colors we need to color all graphs G2,o is lies between 8 and 12.
The upper bound 12 comes from a straightforward induction argument that is based on
Eulers formula for plane graphs. The lower bound 8 is due to the fact that Ks has
thickness-two as Ringel indicated [JT95], In 1974, Sulanke showed that the 9-chromatic
join of K6 and C5 has thickness two. Later, Boutin, Gethner, and Sulanke established
infinitely many 9-chromatic critical graphs of thickness two [BGS08] [GS09],
21


In fact, Ringels Earth-Moon Problem can be generalized to other surfaces and more
than two thickness. It is well-known that Gfj0 is 6t-colorable. However, when t > 3
we do not know if complete graphs have the maximum chromatic number among all
graphs of thickness t. Except, if t >3, then K( thickness t [BW83],
For a given genus g. the thickness t of a graph G is the minimum number of t embeddable
graphs ong for which the union is G. If g is the projective plane, torus, or double-torus,
then the thickness of A), is (n + 5)/6, (// + 4)/6, and(// + 3)/6, respectively [Bei97], Let
B(g) denote the maximum number of vertices of a complete graph that is biembeddable
on the orientable surface g. Eulers formula implies that:
B(g) < [13x|E^£j
Accordingly, Table 2 shows the maximum order B(g) of a complete graph that is
^-biembeddable for g < 3.
Table 2: The functions B(g) for a surface of genus g
g 0 1 2~
B{g) 8 13 14 16
In fact, for any given graph G2>i which can be embedded on the torus with
0(G) = 2, the minimum degree of a vertex is less than or equal to 12, thus
X(G2,i) <13. This comes from a straightforward induction argument that based
on Eulers formula for toroidal graphs with genus g = 1. lacksonand Ringel [JROO]
proved it and Sulanke [Sul05] showed that Ku is biembeddable. Sulanke found 22 graphs
with 13 vertices which can be embedded on the torus and whose complements can also be
embedded on the torus, see figure 3.2 for an example [Sul05],
22


Figure 3.2: Biembedding ofW13
3.1.3 Conclusion
We have studied the coloring problem in detail. We have seen that the Four-Color
theorem proved after a century from it was defined. We have looked at the Heawoods
number and M-pire problem. Moreover, we have seen that 6M colors are sufficient for
all M > 1 [Hea90], We also studied the M-pire special case which is Earth-Moon
problem and its generalization for higher thickness and different surfaces. In summary,
the Earth-Moon problem and its generalization to higher thickness and different
surfaces it has not been solved yet, except for the toroidal graphs on a surface with Euler
characteristic 0 [JROO] [Sul05],
3.2 Visibility Representation Problem
3.2.1 Introduction
Visibility representations (VR) of graphs map vertices to sets in Euclidean space and
express edges as visibility relations between these sets. For graphs G = (V, E) the 3-
dimensional visibility representation by rectangles define as an arrangement of disjoint
rectangles in R3 such that the planes determined by the rectangles are vertical to the
r axis, and the sides of the rectangles are parallel to the x-axes and y axes. A given
graph G is said to be representable if and only if its n vertices can be associated with n
disjoint rectangles parallel to the x and y axes in R3 such that vertex v, and v, are
adjacent in G if and only if their corresponding rectangles R, and A, arez-visible.
23


The current best upper bound on the size of the largest clique Kn that can be
represented by rectangles is 50[BDHS97] and the lower bound is 22[BEL+93], It still a
challenge to narrow the gap between the known upper and lower bound for representation
of Kn by rectangles, thus the question is:
3.2.2 Related Studies
The problem of representing graphs has been studied extensively in the literature as a
result to the large number of applications such as VLSI design, CASE tools, circuit
board layout, and animation problem on 2-dimensions. See [TT86] [DH97] for examples.
Bar-visibility graphs (BVG) in 2-dimensions (also known as bar-representable or s-
visible graphs) were introduced by Luccio, Mazzone, and Wong [LMW87], Wismath
[Wis85], and Tamassia and Tollis [TT86], In this form vertices map to disjoint,
horizontal line segments in the plane, and two vertices are adjacent in the graph if and
only if their corresponding segments are visible in the vertical direction. We can see
that bar-visibility graphs are planar; moreover they have been characterized as those
planar graphs that can be drawn in the plane with all cut vertices on a single face. In
other words, a graph has a bar-visibility representation if and only if it has a planar
embedding such that all cut vertices lie on the external face [Wis85] [TT86], The
question of whether a graph has a bar-visibility layout can be determined in linear time
[KW89],
Rectangle-visibility graph (RVG) in 2-dimensions [BDHS97] is a visibility repre-
sentation in the plane in which the vertices of the graph map to closed rectangles and the
edges are expressed by horizontal or vertical visibility between the rectangles. Two
rectangles are only considered to be visible to one another if there is a nonzero width
horizontal or vertical band of sight between them. Kirkpatrick and Wismath [KW89]
have shown that every planar graph is a rectangle-visibility graph. Hutchinson, Shermer,
and Vince [Bei97] showed that the maximum number of edges e in a rectangle-visibility
graph with n vertices is:
However, thickness-two graphs may have at most 6n~ 12 edges. Dean and Hutchinson
[DH97] established the following bound for bipartite graphs. For n > 4, a bi- partite
rectangle-visibility graph with n vertices has at most 4n 12 edges.
What is the upper bound on the size of the largest clique that
can be represented in 3-dimensions?
\2j
6n 20 if n > 8.
if n < 8,
24


Also, they proved that K5 5 is not a rectangle-visibility graph and complete graph
is the largest complete graph that admits a visibility representation in this repre-
sentation. See figure 3.3.
Figure 3.3: Aghas a rectangle-visibility graph
Recently, interest in finding visibility representation in 3-dimensions has received
considerable attention. A 3-dimensional visibility drawing was introduced as a
generalization of the 2-dimensions visibility drawing. Additional development of bar-
visibility graphs that was studied in [BEL+93] is the class of VR-representable graphs,
which is the representation of vertices by rectangles in 3-dimensions. In this
representation each vertex of the graph maps to a closed rectangle in R3 and edges are
expressed by vertical visibility between rectangles. The rectangles representing vertices
are disjoint, contained in planes vertical to the z-axis, and have sides parallel to the x or
y-axes. Two vertices are adjacent in the graph if and only if their corresponding
rectangles are visible in the z direction.
A lot of papers are focused on the maximum size of a complete graph with 3-
dimensions visibility representation by rectangles. In 1994, Bose, Everett, Fekete, Lubiw,
Meijer, Romanik, Shermer and Whitesides constructed a rectangle visibility
representation of the complete graph K%) based on two main structuring blocks. A K9
can be built by placing K4 and K5 blocks one upon the other as in figure 3.4. The
right edges of the K5 and the bottom edges of the K4 provide the visibility needed to
achieve a K9. A Ajx is constructed as a join of two K9 configurations. One K9 is
placed as the down K9 and the other is placed upon the down K9, The upper K9 is an
exact copy of the down one, but flipped over and rotated 90. The top edges of the
down K9 and the left edges of the up K<, provide the visibility needed to achieve Aj8,
Therefore, by adding the upmost U and downmost D rectangles we have a rectangle
visibility representation of Kliu see figure 3.5 [BEF+94], It was the best known lower
bound of the largest clique that can be represented in 3-dimensions by rectangles.
25


Figure 3.5: Representation of K1{]
Recently, Rote and Zelle [FM99] have found a rectangle-visibility representation of K22
by using simulated annealing algorithm. In [CDHM96] it was shown that no complete
graph with more than 102 vertices has such a representation. This result was then
improved to 55 in [BEL+93] by Fekete, Houle, and Whitesides, they proved that K56
does not have a rectangle-visibility representation. Their proof is based on the analysis
of unimaximal subsequences in sequences of rect- angle coordinates. A sequence xh x2,...
of distinct integers is called unimaximal if it has exactly one local maximum, i.e., for
all /, k with / / k we have */ > min{xh xk}. Eventually, by Stola
[BDHS97][S09] it was reduced to 50, her proof is also based on the study of
unimaximal subsequences in the sequences of rectangle coordinates but she consider each
coordinate dependently.
There are two ways in which two bodies can be considered equal [FHW95]:
(a) Isothetic: they can be made identical by translations only.
(b) Congruent: they can be made identical by translations and rotations.
26


So, two isothetic objects have the same size, while two congruent objects have the same
shape. According to [FHW95], if the vertices are represented by unit squares then the
largest complete graph with this type of representation is K7. See figure 3.6, the six
phases indicate how the seven squares are places on top of each other.

Figure 3.6: Representing K7by unit squares
In summary, Fekete and Meijer [FM99] provided the results for rectangle-visibility
representations for the given number of shapes or sizes. The following Table 3 shows
the best known upper bound for the given number of shapes or sizes; the (min) indicates
the best known lower bound and (max) the best known upper bound.
Regarding complete bipartite graphs, in [BEF+94] the authors found that there is a
rectangle-visibility representations of complete bipartite graphs Km for any m and n ,
where m and n are the size of each partition. In figure 3.7 it is easy to see that any Kmn
admits a rectangle-visibility representations.
Figure 3.7: Any Km n has a rectangle-visibility representation
27


Table 3: Lower and upper bounds for rectangle-visibility representations
Num. of shapes min max Num. of sizes min max
1 12 14 1 7 7
2 18 28 2 12 14
3 20 42 3 18 21
4 20 55 4 20 28
5 20 55 5 20 35
6 22 55 6 20 42
7 20 49
8 20 55
9 20 55
10 20 55
11 22 50
Furthermore, they also proved that there is a rectangle-visibility representation for
complete bipartite graphs that are formed by removing a perfect matching, denoted by
K M, where n is the size of both of the partitions. For proof see [BEF+94],
According to Fekete, Houle, and Whitesides [FHW95], any complete graphs can be
represented by unit discs. See figure 3.8.
Figure 3.8: Any K has a discs-visibility representation
28


3.2.3 Conclusion
We have studied the visibility representations of graphs in 2-dimensions and 3-
dimensions. In 3-dimensions, we have seen that all planar graphs are rectangle-
visibility graph. Complete graphs Kn have rectangle-visibility representation for 11 -
22. Regarding complete bipartite graphs, we have known that Kmjl has a rectangle-
visibility representation for all m and n and K minus a perfect matching admits
rectangle-visibility representation. Finally, we have looked at variant visibility
representations by unit squares and discs as well.
29


4. Optimization Search Methods
The meaning of optimization is finding a parameter in a function that makes a better
solution [DB07], All of appropriate values are possible solutions and the best value is
an optimum solution. To solve optimization problems, optimization algorithms are
used. Categorization of optimization algorithm can be performed by looking at the
nature of the algorithms, and this divides the algorithms into two categories:
deterministic algorithm, and stochastic algorithms [Bla89],
Deterministic algorithms follow a precise procedure, and its path and values of both
design variables and the functions are repeatable. Stochastic algorithms, in general are
called heuristic algorithms. The basic idea for all heuristic algorithms is to choose an
initial solution and then trying to improve this solution by choosing another solution
that belongs to the neighborhood of the current solution. If some requirements are
satisfied, this new solution is accepted to be the new cur- rent solution. All heuristic
methods have a stopping condition, which might be the number of iterations, or
showing that there is no opportunity to improve the current solution anymore [Ski08]
[Por05],
In this chapter, we will discuss and look at four different heuristic search methods:
simulated annealing algorithm and three of the Swarm Intelligence algorithms, which are
ant colony optimization algorithm, cuckoo optimization algorithm, and firefly
algorithm.
4.1 Swarm Intelligence Algorithms (SI)
Swarm intelligence (SI) is based on collective behavior of self-organized systems, natural
examples of SI algorithms: Particle Swarm Optimization (PSO), Ant Colony
Optimization (ACS), Firefly Algorithm (FA), Cuckoo Optimization Algorithm
(COA), Bacteria Foraging (BF), the Artificial Bee Colony (ABC), and so on. It can be
used in controlling robots, predicting social behaviors, enhancing the telecommunication
and computer networks, etc. Actually, the SI can be applied to a variety of fields in
engineering and computer science [PCN10],
In general, a swarm intelligence algorithm for optimization problems works with a
swarm of individuals where each individual creates one solution. Then the solutions are
improved heuristically by using the information about good solutions that have already
been obtained by the swarm. Moreover, researchers have developed multi swarm
versions for particular importance, for example when more than one optimal solution
should be returned. [IMM08],
30


We review some popular algorithms in the field of swarm intelligence for problems of
optimization as follows:
4.1.1 Ant Colony Optimization Algorithm (ACO)
Ant colony optimization is a heuristic algorithm [DBS06], it is one of the most
recent techniques for approximate optimization. It is inspired by adaptation of a
natural system from real ant colonies. In particular, ACO is inspired from the foraging
behavior of ants. When searching for food, ants initially discover the area surrounding
their nest in a random manner. As soon as an ant finds a food source, it evaluates the
quantity and the quality of the food and carries some of it back to the nest. During
the return trip, the ant deposits a chemical pheromone trail on the ground. The
quantity of pheromone deposited will guide other ants to the food source. This
communication between ants via pheromone trails enables them to find shortest paths.
ACO has been initiated by Marco Dorigo [Dor92] which has been successfully applied
to several NP-hard combinatorial problems such as vehicle routing problem, traveling
salesman problem, production scheduling, sequential ordering problem,
telecommunication routing, etc. [GN10],
Figure 4.1 shows the algorithm in simplified model as following steps [Blu05]:
(a) All ants are in the nest. There is no pheromone in the environment.
(b) The foraging starts. In probability, 50% of the ants take the short path (symbolized
by circles), and 50% take the long path to the food source (symbolized by squares).
(c) The ants that have taken the short path have arrived earlier at the food source.
Therefore, when returning, the probability to take again the short path is higher.
(d) The pheromone trail on the short path receives, in probability, a stronger
reinforcement, and the probability to take this path grows. Finally, due to the
evaporation of the pheromone on the long path, the whole colony will, in
probability, use the short path.
The differences between the model and the behavior of real ants is the model consists of a
graph G = (V, E), where V consists of two nodes, namely vs representing the nest of the
ants, and vd representing the food source. Furthermore, E consists of two links, namely c,
and e2, between vs and vd To ex we assign a length of lx, and to e2 a length of l2 such
that l2 > l]. In other words, ex represents the short path between vs and vd and e2
represents the long path. Real ants deposit pheromone on the paths on which they move.
31


Vp**t
(a)
(b)
(O
Figure 4.1: Ants find shortest path between their nest and food sources
Thus, the chemical pheromone trails are modeled as an artificial pheromone value r, for each
of the two links eu i = 1, 2. Such a value indicates the strength of the pheromone trail
on the corresponding path. Each ant Starts from vv and move with probability:
Pi = -Ii where / = 1,2
ri +T2
Obviously, if Ti > t2, the probability of choosing c, is higher, and vice versa. For
returning from vd to yv, an ant uses the same path as it chose to reach vd and it changes
the artificial pheromone value associated to the used edge. Finally, all ants conduct their
return trip and reinforce their chosen path.
Pseudo Code
The basic ant colony optimization algorithm (ACO) can be summarized as the pseudo
code shown in figure 4.2 [DBS06],
Example and Algorithm Trace
(a) Example:
We will clarify the steps of the ACO algorithm by an example to find largest
independent set S of vertices of V in a graph G = (V, E) such that for each
edge (x, y) £ E, either x £ S or y £ S. In figure 4.3 [Ski08] the graph G
is the input, and by applying ACO algorithm we will get the supset S, the output.
The challenge lies not in finding an independent set, but in finding a largest
independent set. Furthermore, finding a maximum independent set is NP-complete
[Ski08],
32


Aiit C olony Optimization Algorithm (ACO)
while {not termenatlon)
Place all ants on the start node S'.
for i=l to n // n the number of ants
fox J7=l to v // v the number of nodes
Choose the next node to visit according to the
probabilistic decision;
If the object solution is met, break;
for j7l to v // each node
Update the pheromone according to local updating
decision;
for A^l to n // each node
Calculate the cost solution from Route [k\ ;
if the best tour from this iteration Is better than the
globally best tour, then set this is the globally best
tour;
end while;
Figure 4.2: Pseudo code of ACO algorithm
ivrri
ni.lli'l
Figure 4.3: Example for Applying ACO Algorithm
The steps will be repeated until termination condition satisfied, which is in this case
the number of iteration. First place all ants on the start vertex (start), let subset S0 =
{start} and initialize pheromone trails. Then, do the following to the n ants:
1- S = S0
2- For j = 1 to the number of vertices
3- Choose the next vertex (next) to visit according to the probability P =~yt-
If {start, next) ^E then S = S+ {next}.
4- Update the amount of pheromone for each node according to the local solution.
5- If the new solution S from this iteration is better than current solution ,S'0 then
replace the current solution with the new one. If the number of iterations is reach to
the maximum, then terminates the algorithm; otherwise go to step 2.
33


After termination we will get the optimal solution for the maximum independent set S
for a given graph G.
(b) Algorithm Trace:
The algorithm trace is for the example. Let the number of iterations = 2 and the
number of ants = 3. All ants will start from the vertex starl, see figure 4.4 (a). Let
S0 = {startj, and the pheromone on each vertex = 0.
For first ant do the following: Initialize S to be SO, then choose the next vertex to
visit by the probability P which is equals to 0 for each move in the first iteration.
The first ant has two choices to move with the same probability as shown in figure
4.4 (b), so let choose one of them as a next, see figure 4.4 (c), but (stare, next) is an
edge in the graph, so nothing will change and choose the next vertex to visit. Figure
4.4 (d) shows that the ant has also two choices to move with the same probability, so
let choose one of them as a next as shown in figure 4.4 (e), (stare, next) is not an edge in
the graph. Thus, S = S + {next} see figure 4.4 (f), and so on until the first ant finish its
tour by reaching the maximum number of vertices. At this point, the first ant give us
the first solution. Next, update the pheromone trail (increment by one) on each vertex
which has been added to the S. Repeat the same steps to the two remaining ants for
the first iteration. After that choose the best solution and replace the current
solution with the best new one and repeat for the second iteration. After
termination we will get the optimal solution for the maximum independent set S for a
given graph G.
34


" I





Figure 4.4: Trace for Applying ACO Algorithm
4.1.2 Cuckoo Optimization Algorithm (COA)
Cuckoo optimization algorithm was developed by Yang and Dep [YD09], It is a
heuristic search algorithm inspired by the cuckoo bird breeding behavior [KC11], The
cuckoo bird lays her eggs in the nest of another host species. The host takes care of the
eggs believing that the eggs are its own. If the host discovers that an egg is not its own,
it may either destroy the egg or the nest and then build a new nest at a different
location. The cuckoo breeding analogy is used for developing new design optimization
algorithm. A generation is represented by a set of host nests. Each nest carries an egg
(solution). The quality of solutions is improved by generating a new solution from an
existing solution by modifying certain characteristics. The new solution is formed by a
random move on the selected solution. If the new solution is found to be superior to
another randomly chosen existing solution then the old solution is replaced with the
new one. Thus, the best solutions in each generation are carried over to the next
generation.
To start the optimization algorithm [Raj 11], each cuckoo starts laying eggs randomly
in some other host birds nests. After all cuckoos eggs are laid in host birds nests, some of
them are detected by host birds and they thrown out of the nest, these eggs have no
chance to grow. Rest of the eggs grow in host nests, hatch and are fed by host birds.
Another interesting point about laid cuckoo eggs is that only one egg in a nest has the
chance to grow. This is because when cuckoo egg hatches
35


and the chicks come out, she throws the host birds own eggs out of the nest. In case
that host birds eggs hatch earlier and cuckoo egg hatches later, cuckoos chick eats
most of the food host bird brings to the nest (because of her 3 times bigger body, she
pushes other chicks and eats more). After couple of days the host birds own chicks die
from hunger and only cuckoo chick remains in the nest. When young cuckoos grow, they
live in their own area and society for some time. But when the time for egg laying
approaches they immigrate to new and better environment with more similarity of eggs
to host birds and also with more food for new youngsters. This environment is
selected as the goal for other cuckoos to immigrate. After that evaluated the new
environment with the previous one and chose the best as a solution. When all cuckoos
immigrated toward goal point, then new egg laying process restarts.
Pseudo Code
The basic cuckoo optimization algorithm (COA) can be summarized as the pseudo code
shown in figure 4.5 [Raj 11],
C'lit'koo Semvli Algorithm (CS)
while (t < maxGeneration) or {stop condition)
Initialize cuckoo with random solution
for i 1 to n //all n cuckoos
Dedicate some eggs and nest to each cuckoo
let cuckoos to lay eggs inside their corresponding nest
kill those eggs that are recognized by host birds
let eggs hatch and chicks grow
limit cuckoos who live in worst nests
find best group and select goal nests
let new cuckoo population immigrate to goal nests
Evaluate new solutions and update current solution
end while:
Figure 4.5: Pseudo code of CS algorithm
Example and Algorithm Trace
(a) Example:
We will explain the steps of the COA algorithm by an example to color the vertices
V in a graph G = (V,E) using the minimum number of colors such that x and y
have different colors for all (x, y) EE. In figure4.6 the graph Q is the input, and
by applying COA algorithm we will get the output colored graph [Ski08],
36


Figure 4.6: Example for Applying COA Algorithm
To apply the COA algorithm, we repeat the following steps until number of vertices
that properly colored reach to the maximum.
1- First initialize solution S with a random solution. Then, do the next steps to n
cuckoos.
2- Give colors to a cuckoo.
3- Let the cuckoo to color vertices.
4- uncolored the vertices that are recognized not properly colored.
5- Count the vertices that properly colored, and reduce the number of cuckoos who
improperly colored vertices.
6- Evaluate new solution and update current solution S if the new solution is better
than current. If the number of iterations is reach to the maximum, then
terminates the algorithm; otherwise go to step 1.
After the algorithm terminates we will have an output as a graph G properly
colored with minimum number of colors.
(b) Algorithm Trace:
The algorithm trace is for the example. The steps will be repeated until the
number of vertices that are properly colored reach 10. First initialize solution S with a
random solution as shown in figure 4.7 (a). Let the number of cuckoos = 2.
For the four cuckoos do the following: Give first color to the cuckoos and let the first
one to color the vertices, see figure 4.7 (b). Then uncolored the vertices that are
recognized not properly colored and count the number of vertices that are properly
colored, which are (3 vertices) the first solution, see figure 4.7 (c). Then
37


let the second cockoo to color vertices with the same color randomly as shown in
figure 4.7 (d). Then uncolored the vertices that are recognized not properly colored
and count the number of vertices that are properly colored, which are (3 vertices) the
second solution, see figure 4.7 (e). After that reduce the number of cuckoos who
improperly colored largest number of the vertices (which is the second cuckoo). Thus,
the number of cuckoo is one. Evaluate new solutions and update current solution as
shown in figure 4.7 (f). Repeat the same steps with the second, third, fourth, ..., n
colors, until the number of vertices that are properly colored reach 10.
Figure 4.7: Trace for Applying COA Algorithm
4.1.3 Firefly Algorithm (FA)
The firefly algorithm is a heuristic algorithm [Yan09], it inspired by the flashing
behavior of fireflies. The fundamental purpose of fireflys flash is to act as a signal
system to attract other fireflies. Thus, each fireflys movement is based on absorption of
the others. The flashing light can be formulated in such a way which makes it possible to
invent new optimization algorithms [CM12], For simplicity, Yang formulated these
flashing characteristics as the following rules:
1. All fireflies are unisex, so that one firefly will be attracted to all other fireflies
regardless of their sex.
38


2. Attractiveness is relative to fireflys brightness, so for any two fireflies, the less
bright firefly will be attracted to the brighter one and move toward it. However,
the brightness can decrease as their distance increases. If there is no brighter one
than a given firefly, it will move randomly.
3. The brightness of a firefly is determined by the objective function.
Pseudo Code
Based on previous three rules, the basic firefly algorithm (FA) can be summarized as the
pseudo code shown in figure 4.8 [Yan09],
Firefly Algorithm (FA)
Objective function £ (x) x = (xi, . x^J7
Generate initial population of fireflies xt \i =1,2, . n)
Light intensity Ii at xi is determined by f {x,)
while {t < MaxGeneraticri)
for i 1 t n //all n fireflies
for 7 = 1 to i //all n fireflies
If (I-* > I<) Move firefly i towards j;
Evaluate new solutions and update light intensity for
Attractiveness
end for j
end for i
Rank the fireflies and find the current global best
end while
Figure 4.8: Pseudo code of FA algorithm
Example and Algorithm Trace
(a) Example:
We will demonstrate the steps of the FA algorithm by an example to Color the edges E
in a graph G = (V, E) using the minimum number of colors such that no two same
color edges share a common vertex. In figure 4.9 the graph G is the input, and by
applying FA algorithm we will get the output colored graph [Ski08],
39


Figure 4.9: Example for Applying FA Algorithms
This algorithm starts with generating initial population of n fireflies (vertices).
Determining the light intensity I to each firefly by objective function, which is the
vertex degree in this example. Repeat the algorithm until t reaches the number of edges
e. Initialize color c with first color and then do the next steps:
1- For i = 1 to n (number of fireflies) do
2- For j = i + 1 to n do
3- Test if (/, /) is an edge in a given graph then
4- If the Ij > Ij and firefly i have not an edge colored with c, then color the edge
between i and/
5- Evaluate new solution and update light intensity by decreasing It by one.
After finishing the above steps for one iteration increment c by one and repeat until
the algorithm terminates. By the end we will have a graph G properly colored.
(b) Algorithm Trace:
The algorithm trace is for the example. First we are generating initial population of
n firefly which is 8 vertices. Then we are determining the light intensity I to each
firefly which is the vertex degree. All the vertices have degree equals to 3 except the
vertex in the middle has degree equals to 7, see figure 4.10 (a). The steps will be
repeated until the number of edges reaches to the maximum which is 14. Next,
initialize color c with first color and do the next steps to 8 fireflies:
For firefly i = 1 test if there is an edge between it and the firefly j then compare the
degree, see figure 4.10 (b). If the degree of vertex j is greater than or equals the degree
of vertex i and vertex i has not an edge colored with first color, then color the edge
between i andJ as shows in figure 4.10 (c).
40


Then decrement the light intensity to the firefly i by one, see figure 4.10 (d). Do the
same steps with all j fireflies. After finishing with first firefly do the same steps for
firefly / = 2 as showing in figure 4.10 (e and f). Repeat these steps to all i fireflies, then
after the first iteration set color c with the second color and test the stop condition if
it satisfied then the algorithm terminates; otherwise another iteration will takes place.
Figure 4.10: Trace for Applying FA Algorithms
4.2 Simulated Annealing Algorithm (SA)
Simulated annealing is a heuristic algorithm, and it was developed by Kirkpatrick, Gelatt
and Vecchi [KGV88], Simulated annealing inspired from the physical process of cooling
molten materials down to the solid state. Theoretically, the energy state of a system is
described by the energy state of each particle constituting it. The energy state jumps
randomly, with such transitions managed by the temperature of the system. The
transition probability /J(e,, e,,7 ) from energy e, to e, at temperature T is given by:
Pfa, ?/, T) = e 1^fL
Where ks is a constant[Ski08],
41


There are two major processes that the simulated annealing algorithm must go
through. First, for each temperature the algorithm runs through a number of iteration.
The number of iteration is determined by the programmer. In each iterate the inputs
are randomized and once the number of iteration has been completed, the temperature can
be lowered. Second, if the temperature is lowered, it is decided whether or not the
temperature has reached the lowest temperature allowed. If the temperature is not
lower than the lowest temperature, then the temperature is lowered and another
iteration of randomizations will take place. If the temperature is lower than the lowest
temperature, the simulated annealing algorithm terminates.
Pseudo Code
The basic simulated annealing algorithm (SA) can be summarized as the pseudo code
shown in figure 4.11.
Simulated Annealing Algorithm (SA)
Create initial solution S
Initialize temperature t
repeat
for i = 1 to i ierva tlon-length do
Generate a random transition from S to Si
If {C{S) 2 C{SJ) then S = Si
else if -itism }j/{*-£) ;> jrandaialO, 1)) then S = Sa
Reduce temperature t
until {no change in Cf.?))
Return S
Figure 4.11: Pseudo code of SA algorithm
Example and Algorithm Trace
(a) Example:
We will clarify the steps of the SA algorithm by an example to find largest clique
S in a graph G = (V,E) such that for all x, y E S, (x, y) E E. In figure 4.12
[Ski08] the graph G is the input, and by applying SA algorithm we will get the
largest clique S, the output. The challenge lies not in finding a clique, but in finding
a largest clique. Furthermore, finding a maximum clique is NP-complete [Ski08],
42


INl'111
u'ni'c
Figure 4.12: Example for Applying SA Algorithms
First, create initial solution S() which is a subgraph of G with e = "(" l; edges.
Initialize temperature t. The steps will be repeated until the temperature reaches zero or
there is no better solution (no subgraph larger than S with e = 2] > edges). Then,
do the follows for i=n (number of vertices) iterations:
1- Generate random solution S in the neighborhood of solution S0 and compares their
number of nodes and edges.
2- If S is better than S0, then S is accepted as a new solution; otherwise, S is accepted
with a probability P. The higher the temperature, the more likely it is to accept
worse solutions.
3- Decrease the temperature. Next, decided whether or not the temperature has reached
the lowest temperature allowed (zero). If the temperature is not zero another
iteration will take place; otherwise, the algorithm terminates.
After termination we might get the solution for the maximum clique S for a given
graph G if it found.
(b) Algorithm Trace:
The algorithm trace is for the example. First we generate initial random solu-
tion No which is a subgraph of G with the n vertices and e = "("2 ].
See figure 4.13 (a) for the solution S0 which is a subgraph of G with the 3 vertices and
3 edges (K3). Then we initialize the temperature to 100. The steps will be repeated
until the temperature reaches zero or there is no change on the solution.
For the number of vertices 17 do the following: Generate a random solution S from
the neighborhood of the solution S0 and compares their number of vertices and edges, see
figure 4.13 (b). If solution S is better than solution S0 then S is accepted as anew
solution; otherwise S is accepted with the probability P which is depends on the
43


difference between S and S() and the temperature. The higher the temperature, the more
likely to accept worse solution. Thus, solution S is as same as solution S0 and we
accepted as anew solution as shown in figure 4.13 (c). After 17 iterations decrease the
temperature by 0.96 and another iteration will take place until the temperature
reaches zero or there is no change it the solution S.
44


5. Research Methodology
This research was conducted in order to combine heuristic search methods with graph
theoretic methods to solve two classical problems in graph theory (Graph coloring
problem and certain visibility representations problem for graphs). First, they are used
to generate graphs with specific chromatic numbers and finding its thickness, and vice
versa. The goal of creating and exploring these graphs is to discover more about the
bounds of the chromatic number for the Earth-Moon problem and its generalization to
higher thickness and different surfaces. Second, they are used to represent the problem of
visibility in graph form and to discover more about the bounds of the largest size of
complete graphs that have a rectangles visibility representation in 3-dimensions. In order
to achieve these research goals, we chose the three following techniques:
5.1 First Technique
We used a technique inspired by the next proposition. To that end, let G represent the
complement of G.
Proposition 1 [GS09] Suppose G is a graphwith n vertices. If G is Km-free,
then X(G)>[^|
Proof Given a graph G, the vertices being colored with the same color form an
independent set, which in turn form a complete subgraphs in the complement.
Hence, if G dose not contain Km then at most m l vertices of G can be colored
with the same color, which leads:
x(G)>r^rl

In this technique, we start with a graph G of known thickness t on genus g and find its
chromatic number by eliminating the number of Km subgraphs of the com-
plement of the union of t subgraphs until it reaches zero.
This technique is applied by using three different optimization algorithms
(Ant Colony Optimization, Firefly Algorithm, and Simulated Annealing).
45


Implementation
In order to make the complement G of G K m-free and produce -critical
graph with n vertices, we performed the following steps for both the Ant Colony
Optimization and Firefly Algorithms:
1. Generate t triangulations on genus g by using Sulankes Triangulations
software.
2. Find an initial solution by randomly flipping the diagonals of quadrilaterals in the
triangulation.
3. Count the number of Km subgraphs on the complement of the union of t
triangulations.
4. Improve the current solution by randomly flipping the diagonals of quadri-
laterals of the triangulations until the complement of their union has no Km.
5. Test each remaining edge of the triangulations one at a time, remove any edge
which leaves the complement of the union with no Km.
6. Form the union G of the t triangulations.
7. If any vertex of G has degree less than / 1 then reject G; otherwise,
8. Check that the graph G is -V_j / -chromatic by using Dharwadkers software
The Vertex Coloring Algorithm [Dhal 1],
9. Test each remaining edge of G one at a time, remove any edge which leaves
G f-. /chromatic.
m 1
10. If G has less than n vertices then reject G; otherwise,
11. G is now f-. /critical with n vertices.
m 1
However, in the simulated annealing algorithm we performed exactly the same steps
with a small difference. In step four we find a new random solution in the
neighborhood of the current solution and compares their values. If the new solution is
better than the current, then the new solution is accepted as the new current solution;
otherwise the new solution is accepted with the probability that depends on the difference
in the objective functions and on the temperature. The higher the temperature, the more
likely it is to accept worse-local solutions. While in the previous algorithms we just
accepted the better-local solutions.
46


5.2
Second Technique
We used a technique inspired by the next propositions.
Proposition 2 Let G be any graph. Then /(G) > K(G')'
a(G)
Proof Given a ^-coloring of G, the vertices being colored with the same color form
an independent set. Let G be a graph with n vertices and c a ^-coloring of
G. We define V, = jv c(v) = ij for i = 0,1,..., k.
Each V, is an independent set. Let a(G) be the independence number of G, we have V, <
a(G). Since
n = \V(G)\ = \Vi\ + \V2\ + + \Vk\ we have:
X(G) >\
V(G) |
WgT7

Proposition 3 Suppose G is a graph with Euler characteristic e on n vertices and e
edges, then the thickness of G, (G) > i3(ne_£)i
Proof It is well-known that a planar graph on genus g with n >3 can have no more
than 3(n- e) edges. A maximal planar graph is one to which no edges can
be added without destroying planarity. Given a non-planar graph G, a maximal planar
subgraph H of G is a subgraph of G to which no edges of G H can be
added without destroying planarity. A maximum planar subgraph is one which has
maximum number of edges among all planar subgraphs of G. The thickness, 0(G), of a
graph G is the minimum number of planar subgraphs whose union is G, which leads:
etc) > r^i c
We start with a graph G on genus g of known higher chromatic number according to
Proposition 2 and find its thickness by finding subgraphs whose union is G, each
subgraph has enough edges to have thickness t illustrated in Proposition 3. This
technique applied using cuckoo optimization algorithm, a general randomized heuristic
approach for finding good solutions for optimization problem.
47


Implementation
In this implementation, we found the decomposition of a given graph G into t
subgraphs which can be embedded on an orientable genus g, by the following steps:
1. Generate a graph G.
2. Check that the graph G is v<<'> -chromatic by using Dharwadkers software
a(G)
The Vertex Coloring Algorithm [Dhal 1],
3. Random planar graphs (triangulations) of the selected genus are generated with at
most 3(n s) edges and at most 2(n e) faces, and do not contain any subdivision
ofifj or KAAS as illustrated in Kuratowskis Theorem 4.
4. Randomly interchanging diagonal to minimize the number of edges not in the
triangulations but are in the graph being decomposed until there is no edge that
dose not belongs to any triangulations.
5. Now, G is decomposed into t subgraphs.
5.3 Third Technique
We consider sequences of n rectangles lying parallel to the x, y-plane in R3. The sequence
is valid if its associated visibility graph is Kn. Each rectangle R in a valid sequence
can be describe in terms of the perpendicular distances from common point O to each of
its sides. Instead of giving the x, y-coordi nates of R, we describe R as a 4-tuple (Er, Nr,
Wr, Sr) whose coordinates given respectively, the distances from 0 ER to the east, north,
west and south sides of R.
We assume that each coordinate value of each of the n rectangles is a positive integer in
range [0, ri\ without changing the visibility relationships among the rectangles. Suppose
that two rectangles in a valid sequence, A = (Ea, Na, Wa, Sa) and B=(Eb, Nb, Wb, Sb). the
intersection of their projections onto the x, y-plane denoted by A fl B. Then A B
contains 0, and the coordinates of A B are EAnB = min{Ea, Eb}, NA = min/Ay,
Ay/, WAcb = min/if,, Wb} and SA = min {Sa, Sbj. We say that a corner of A B is
free if it is not covered by any of the projections of rectangles occurring between^ and B
in the sequence [BEF+94],
In summary, the rectangle A and B can see each other if and only if one of the
following free corner conditions F C holds for all the rectangles R between A and B
[BEL+93]:
48


FCne(A, B) northeast is free, i.e. (Er < min {Ea, Eb} or Nr < min {Na, Nb})
FCW(A, B) northwest is free, i.e. (Nr < min {Na, Nb} or W< min {Wa, Wb})
FCSW(A, B) southwest is free, i.e. (Wr < min (Wa, Wb} or Sr < min {Sa, Sb})
FCse(A,B) southeast is free, i.e. (Sr < min /.S,, Sb} or Er < min {Ea, Ebj)
The representation of Kn is found using three different optimization algorithms (Ant
Colony Optimization, Cuckoo Optimization Algorithm, and Firefly Algorithm).
Implementation
For given //, the algorithms try to find a realization of Kn as follows:
1. Generate an initial random solution.
2. Count the number of rectangles needed to be removed from the collection in
order that each rectangle can see the others (objective function). That is
determined in a straightforward manner from the free corner conditions FC.
3. Improve the current solution by randomly selecting two rectangles and swap- ping
one of the four coordinates until the objective function reaches zero.
4. Now, Kn is representable with rectangles in 3-dimensions.
49


6. Findings and Discussions
In this chapter, we conclude this study by showing our results and discussing some
recommendations for future work.
6.1 Results of the Study
In this study, we have focused on two problems relevant to the combinatorial problems
in graph theory:
(i) Graph coloring problem
(ii) Visibility representation for graph in 3-dimensions
The aim of this thesis was to improve the well-known upper bound for the chromatic
number of any genus g thickness 1 orientable graph, and the upper bound on the size of
the largest clique that can be represented in 3-dimensions by means of the heuristic search
methods. In order to do that we tried to answer several questions (see research
questions). The results were as follow:
First: Does there exist a graph G2,o with /(G2}o) = 10, 11, or 12?
To answer the first question, we have to show that the upper bound of the chromatic
number for any genus 0 and thickness two graph is 12, then we apply the first technique,
which is inspired by Proposition 1.
The Euler characteristic for orientable surfaces g is e(g) = 2-2g, so e(0) = 2. The M
-pire chromatic number for any surface with Euler characteristic e is at most
[Hea90][JR85]:
V____________
/(M -pire, e) < 6M+i+ (6A/+i)2-24e
2
V______________
Hence,/(2-pire, 2) < (6x2)+i+ (6x2+i)2-24(2)
2
Corollary 2
If 23 <\V(G) <24andG is K^-free, then '/(G) > 12.
If 34 <\V(G)\ <36andG is K^-free, then/(G) >12.
If 45 <\V(G) <48 andG is K^-free, then/(G) > 12.
50


For (Km, n) £{(K3, 23), (K4, 34), (K5, 45)/. we have not yet succeeded on finding
12-chromatic thickness two graph on the sphere. The minimum number of com-
plete graphs Km in the complements were 93, 69, and 162, respectively by applied
ACO, FA, and SA algorithms.
Corollary 3
If 21 <\V(G) <22andG is K^-free, then '/(G) >11.
If 21 <\V(G) <33 andG is K4-free, then/(G) >11.
If 41 <\V(G)\<44andG is K5-free, then/(G) >11.
For (Km, n) £ {(K3, 21), (K4, 31), (K5, 41)j, we have not yet succeeded on finding 11-
chromatic thickness two graph on the sphere. The minimum number of complete graphs
Km in the complements were 35, 106, and 140, respectively by applied ACO, FA, and
SA algorithms.
Corollary 4
If 19 <\V(G)\ <20andG is K^-free, then/(G) >10.
If 28 <\V(G)| <30andG is K4-free, then/(G) >10.
If 37 <\V(G) <40andG is K^-free, then/(G) >10.
For (Km, n) £{(K3, 19), (K4, 28), (K5, 37)}, we have not yet succeeded on finding
10-chromatic thickness two graph on the sphere. The minimum number of com-
plete graphs Km in the complements were 5, 71, and 43, respectively by applied
ACO, FA, and SA algorithms.
Corollary 5
If 17 <\V(G) <18 andG is K^-free, then/(G) >9.
If 25 <\V(G) <27andG is K4-free, then/(G) >9.
If 33 <\V(G) <36andG is Kfree, then/(G) >9.
51


For (Km, n) £ {(K3, 17), (K4, 25), (K5, 33)}, we have succeeded on finding 9-
chromatic thickness two graph on the sphere. See Appendix B, it contains 90
new 9-critical graphs. The graphs were selected from a catalogue of hundreds found,
where the candidates were chosen for the uniqueness of degree sequences. Figure 6.1 shows
an example of new 9-critical thickness two graph on 17 vertices whose complement is
ATj-free, figure 6.2 shows an example of new 9-critical thickness two graph on 25
vertices whose complement is A4-free, and figure 6.3 is anew 9-critical thickness two
graph on33 vertices whose complement is A5-free.
Figure 6.1: New9-Critical Graph whose Complement is A",-free
52


i )egree Sequence
[13. 12. 12. II. 11, ] 1.10. 9.9, 9.9, 9. 9, 9. 9, 9. 9. 8, 8. 8. 8. 8. 8. 8,8}
Figure 6.2: New9-Critical Graph whose complement is AT4-free
Degree Sequence
[13.11. 11.11. 10. 10. 10. 10. 10.10,9.9.9,9.9.9.9.9.9.9.9.9.9.9.9. 9.9. 9.9. 9.
8.8.8}
Figure 6.3: New9-Critical Graph whose complement is AVfree
Second: Does there exist a graph G3fi with /(G30) = 17 or 18?
To answer the second question, we have to show that the upper bound of chromatic
number for any genus 0 and thickness three graph is 18, then we apply the first
53


Hence, /(3-pire, 2) <
X(M -pire, e) <
V_______
6M+1+ (6M+1)2
2
V_______________
(6 x 3 )+1 + (6 x 3+1)224(2)
2
<18.
24e
Corollary 6
If 35 <\V(G) <36andG is K^-free, then '/(G) >18.
If 52 <|F(G) <54andG is K4-free, then/(G) >18.
// 69 < V(G) <12 and G is K5-free, then/(G) >18.
For (Km, n) £{(K3, 35), (K4, 52), (K5, 69)}, we have not yet succeeded on finding
18-chromatic thickness three graph on the sphere. The minimum number of com-
plete graphs Km so far in the complements were 371, 606, and 754, respectively by
applied ACO, FA, and SA algorithms.
Corollary 7
If 33 <\V(G) <34andG is K^-free, then/(G) > 17.
If 49 <\V(G)\ <51 andG is K4-free, then/(G) >17.
Jf 65 <\V(G)\ <68 andG is K^-free, then/(G) > 17.
For (Km, n) £ {(K3, 33), (K4, 49), (K5, 65)/. we have not yet succeeded on finding 17-
chromatic thickness three graph on the sphere. The minimum number of complete
graphs Km in the complements were 307, 332, and 696, respectively by applied ACO,
FA, and SA algorithms.
Third: Does there exist a graph G40 with /(G40) = 23 or 24?
To answer the third question, we have to show that the upper bound of chromatic number
for any genus 0 and thickness four graph is 24, then we apply the first technique,
which is inspired by Proposition 1.
The Euler characteristic for orientable surfaces g is fg) = 2-2 g, so e(0) = 2. The AT-pire
chromatic number for any surface with Euler characteristic e is at most:
54


X(M -pire, e) < 6M+i+ (6A/+i)2-24e
Hence,/(4-pire, 2) < (6x4)+i+ (6x4+i)2-24(2) ^
2
Corollary 8
If 41 <\V(G) <48 andG is K^-free, thenx(G) >24.
If IQ <\V (G) <72 and G is K4-free, thenx(G) >24.
If 93 < |F(G) <96andG is K^-free, thenx(G) >24.
For (Km, n) £ {(K3, 47), (K4, 70), (K5, 93)/. we have not yet succeeded on finding 24-
chromatic thickness four graph on the sphere. The minimum number of complete
graphs Km in the complements were 236, 351, and 880, respectively by applied ACO,
FA, and SA algorithms.
Corollary 9
If 45 <\V(G) <46andG is K^-free, thenx(G) >23.
If 61 <\V(G) <69andG is K4-free, thenx(G) >23.
If 89 <\V(f})\ <92andG is K^-free, thenx(G) >23.
For (Km, n) £ {(K3, 45), (K4, 67), (K5, 89)j, we have not yet succeeded on finding 23-
chromatic thickness four graph on the sphere. The minimum number of complete
graphs Km in the complements were 198, 359, and 459, respectively by applied ACO,
FA, and SA algorithms.
Fourth: Does there exist a graph G31 with /(G31) = 19?
To answer the fourth question, we have to show that the upper bound of chromatic
number for any genus 1 and thickness three graph is 19. Then we apply the second
technique, which is inspired by Proposition 2 and Proposition 3.
The Euler characteristic for orientable surfaces g is e(g) = 2-2g, so e(l) = 0. The
M-pire chromatic number for any surface with Euler characteristic e is at most:
55


Hence, /(3-pire, 0) <
X(M -pire, e) <
V_____________
6M+1+ (6M+ l)2-24e
2
V_______________
(6 x3)+l+ (6 x 3+1)224(0)
2
<19.
The graph Kw which has 171 edges is the union of three graphs each of which can be
embedded on genus 1. Since the graph that has n vertices has at most 3(n-s) edges can be
embedded on a surface with Euler characteristic e, then the graph KV) can be embedded
on the torus (genus 1) which has 19 vertices has at most 57 edges. Thus, KV) is a graph
with/(G3,i) = 19.
By applying second technique and the Sulankes Triangulations software, we found
the decomposition of G3,i = KV) into three subgraphs (G3 = Gi u G2 u G3) which can be
embedded on genus 1 as follows:
Figure 6.4: Decomposition of^i9
Fifth: Does there exist a graph G4i with /(G4i) =25?
To answer the fifth question, we have to show that the upper bound of chromatic
number for any genus 1 and thickness four graph is 25. Then we apply the second
technique, which is inspired by Proposition 2 and Proposition 3.
The Euler characteristic for orientable surfaces g is e(g) = 2-2g, so e(l) = 0. The
M-pire chromatic number for any surface with Euler characteristic e is at most:
X(M -pire, e) <
V__________
6M+1+ (6M+\ )2 24e
2
Hence, /(4-pire, 0) <
V_______________
(6 x4)+l+ (6x4+l)2-24(0)
2
<25.
56


The graph K25 which has 300 edges is the union of four subgraphs each of which can be
embedded on genus 1. Since the graph that has n vertices has at most
3(n~e) edges can be embedded on a surface with Euler characteristic e, then the
graph K25 can be embedded on the torus (genus 1) which has 25 vertices has at
most 75 edges. Thus, K25 is a graph with /(G4,i) = 25.
By applying second technique and the Sulankes Triangulations software, we have
not succeeded on finding that K25 has thickness four. There was just one edge that does
not belongs to any subgraphs. The graph K25 has at most thickness five as shown in figure
6.5.
Figure 6.5: Decomposition of^5
But by applying the same methods, we found the decomposition of G4,i = K24, whose
'/(K24) = 24 into four subgraphs (G4,i = Gi u G2 u G3 r G4) which can be embedded on
genus 1 as follows:
57


Figure 6.6: Decomposition of^24
Sixth: Does there exist a graph G2,2 with /(G22) = 14?
To answer the sixth question, we have to show that the upper bound of chromatic
number for any genus 2 and thickness two graph is 14. Then we apply the second
technique, which is inspired by proposition 2 and proposition 3.
The Euler characteristic for orientable surfaces g is s(g) = 2- 2g, so e(2) = -2. The M
-pire chromatic number for any surface with Euler characteristic e is at most:
V___________
X(M-pire, e) < 6M+i+ (6m+ \y-we
2
V_______________
Hence,/(2-pire,-2) < (6x2)+i+ (6x2+i)2-24(-2)
2 -
The graph Ku which has 91 edges is the union of two graphs each of which can be
embedded on genus 2. Since the graph that has n vertices has at most 3(n-s) edges can be
embedded on a surface with Euler characteristic e, then the graph K]4 can be embedded
on the double-torus (genus 2) which has 14 vertices has at most 48 edges. Thus, Ku is a
graph with/(G2j2) = 14.
By applying second technique and the Sulankes Triangulations software, we found
the decomposition of G2 2 = Ku into two subgraphs (G2j2 = Gi u G2) which can be
embedded on genus 2 as follows:
58


1. Gi Triangulation:
0: 2 1310
1: 263 118 5 101374
2: 0 10978 116 1 4 13
3: 1 65 7 139 11
4: 1 79 132
5: 1 873 6 129 10
6: 1 2 119 1253
7: 1 133 5 8294
8: 111275
9: 2 105 126 113 1347
10: 0 13 1 5 92
11: 1 3 9628
12: 5 69
13: 02493 7 1 10
2. G2 Triangulation:
0: 1 12764 11 5 3 8 9
1: 098 12
2: 3 5 12
3: 0 52 121048
4: 06 125 83 1011
5: 0 11 1384 1223
6: 07 108 13 124
7: 0 1211 106
8: 03 4 5 136 10121 9
9: 0 8 1
10: 3 12867 114
11: 04 107 12135
12: 0 1 8 103 2 546 13 117
13: 5 11 1268
Seventh: Does there exist a graph G3>2 with /(G3 2) =20?
To answer the seventh question, we have to show that the upper bound of chro-
matic number for any genus 2 and thickness three graph is 20. Then we apply the second
technique, which is inspired by Proposition 2 and Proposition 3.
The Euler characteristic for orientable surfaces g is e(g) =2-2g, so e(2) = -2. The M
-pire chromatic number for any surface with Euler characteristic e is at most:
59


Hence, /(3-pire, -2) <
X(M -pire, e) < 6M+i+ (6A/+i)2-24e
V_______________
(6x3)+!+ (6 x 3+1)2 24( 2) <2Q
The graph K20 which has 190 edges is the union of three graphs each of which can be
embedded on genus 2. Since the graph that has n vertices has at most 3(n-s) edges can be
embedded on a surface with Euler characteristic e, then the graph K2o can be embedded
on the double-torus (genus 2) which has 20 vertices has at most 66 edges. Thus, K20 is a
graph with/(G3>2) =20.
By second technique and Sulankes Triangulations software, we found the de-
composition of G3j2 = K20 into three subgraphs (G3 2 = Gi r G2 r G3) which can be
embedded on genus 2 as follows:
1. Gi Triangulation:
0: 6 14177 1913 18 11 16108 9
1: 69 197 11
2: 4 191817
3: 6 1714
4: 2 176 11 19
5: 11 1217
6: 09 1 114 173 14
7: 0 171298 101611 1 19
8: 0 1079
9: 0 87 1215 191 6
10: 0 1678
11: 0 18 125 171946 1 7 16
12: 5 11 18191597 17
13: 0 191718
14: 0 63 17
15: 9 1219
16: 0 117 10
17: 0 143 642 1813 1911 5 127
18: 0 13 172 191211
19: 07 1 9 15 121824 11 1713
2. G2 Triangulation:
0: 1 2 15
1: 0 15 171613 10183 142
60


2: 0 1 1495 73 16126 1015
3: 1 18 1627 5 10198 15 11 14
4: 9 1618
5: 29 18 103 7
6: 2 1210
7: 2 5 3
8: 3 1912161715
9: 2 14164 185
10: 1 13 1526 12193 5 18
11: 3 1514
12: 2 168 19106
13: 1 161510
14: 1 3 11 15 1692
15: 02 1013 1614113 8 171
16: 1 178 1223 1849 1415 13
17: 1 158 16
18: 1 105 94 163
19: 3 10128
3. G3 Triangulation:
0: 3 1254
1: 4 8 5 12
2: 8 1311
3: 04 179 13 12
4: 0 5 157 138 1 121410173
5: 0 121 86 16191413 154
6: 5 8 18 15 137 19 16
7: 4 15 1814196 13
8: 1 4 132 11 10141865
9: 3 171011 13
10: 4 148 119 17
11: 2 139 108
12: 0 3 13 144 1 5
13: 2 8476 155 14123 9 11
14: 4 12135 197 188 10
15: 4 5 136 187
16: 5 6 19
17: 3 4 109
18: 6 8 147 15
19: 5 1667 14
61


Eighth: Does there exist a graph G42 with /(G4>2) =26?
To answer the eighth question, we have to show that the upper bound of chromatic
number for any genus 2 and thickness four graph is 26. Then we apply the second
technique, which is inspired by Proposition 2 and Proposition 3.
The Euler characteristic for orientable surfaces g is e(g) =2-2g, so e(2) = -2. The M
-pire chromatic number for any surface with Euler characteristic e is at most:
V___________
X(M -pire, e) < m+\+ csM+i f-iAe
2
V_______________
Hence,/(4-pire,-2) < (6x4)+i+ (6x4+i)2-24(-2)
The graph K26 which has 325 edges is the union of three graphs each of which can be
embedded on genus 2. Since the graph that has n vertices has at most 3{n~e) edges can be
embedded on a surface with Euler characteristic e, then the graph K2(> can be embedded
on the double-torus (genus 2) which has 26 vertices has at most 84 edges. Thus, K26 is a
graph with/(G4j2) =26.
By second technique and Sulankes Triangulations software, we found the de-
composition of G4j2 = K26 into four subgraphs (G42 = G\ uG2U G3 u G4) which can be
embedded on genus 2 as follows:
1. Gi Triangulation:
0: 1 20 9 15 2413 4 3 25
1: 0253 7 1920
2: 172022
3: 04 162220197 1 25
4: 0 13 122014163
5: 13 1614
6: 13 23 17
7: 1 3 19
8: 9 221621
9: 0 2013 11228 21241715
10: 13 142017241523
11: 9 1322
12: 4 13 1820
13: 024165 1410236 172211 92018 124
14: 4 201013 5 16
15: 0 9 1723 1024
16: 3 4 145 132421 822
62


17: 2 2213 6 23 15 9 241020
18: 121320
19: 1 73 20
20: 0 1 19 3 222 1710144 1218 13 9
21: 8 16249
22: 2203 168 9 11 13 17
23: 6 13 1015 17
24: 0 15 1017921 1613
25: 03 1
2. G2 Triangulation:
0: 2 11 1610672214
1: 5 68
2: 0 1418242213257 1611
3: 69 122418 14
4: 5 1917723 2215
5: 1 8 12194 1522247206
6: 0 101693 148 1 5207
7: 0620524128 14234 1716225 22
8: 1 6 147 125
9: 3 6 1612
10: 0 166
11: 02 16
12: 3 9 16195 8724
13: 2 2225
14: 02221237863 182
15: 4225
16: 0 1127 17191296 10
17: 4 19167
18: 2 143 24
19: 45 121617
20: 5 76
21: 142223
22: 0 7 25 13 2 24 5 15 4 23 21 14
23: 47 142122
24: 2 183 127 5 22
25: 2 13227
3. G3 Triangulation:
0: 12 21 18 23
63


1: 3 1321 17
2: 4 6 1223 8 10
3: 1 1711 108235 9 13
4: 2 109 186
5: 3 23 9
6: 24 1822192425 15 12
7: 9 1021 13
8: 2 23 3 10
9: 3 523 25 16184 107 13
10: 2 83 11 1921794
11: 3 1725 1910
12: 02326 1525 141721
13: 1 3 9721
14: 1225 17
15: 6 25 12
16: 9 25 2023 18
17: 1 21 121425 113
18: 021 1922649 1623
19: 6221821 101125 2324
20: 1625 2423
21: 0 12171 137 101918
22: 61819
23: 0 18 162024192595 3 82 12
24: 6 1923 2025
25: 6242016923 1911 17141215
4. G4 Triangulation:
0: 5 178 19
1: 29 1424423 11 12221018 1615
2: 1 153 215 199
3: 2 1521
4: 1 24 8 25216 1123
5: 0 1922125 101211 18 17
6: 421 11
7: 11 15 18
8: 0 171825424112015 13 19
9: 1 2 1914
10: 1 22125 25 18
11: 1 2346212082414157 185 12
12: 1 115 1022
13: 8 15 19
14: 1 9 19151124
64


15 1 16187 11 14 191382021 3 2
16 1 18 15
17 0 5 188
18 1 10258 175 1 1 7 15 16
19 0 8 13 15 149 2 5
20 8 1121 15
21 2 3 1520116 4 25 5
22 1 1210
23 1 4 11
24 1 141184
25 4 8 18 105 21
Ninth: Does the 10-chromatic {4,4,4,4,3}-inflated C5 have thickness two?
In order to answer the ninth question, we have applied the second technique. In this
technique we started with (4,4,4,4,3j-inflated C5 on genus 0, which is 10-
chromatic graph with 19 vertices and 99 edges, by (Definition 9).
Unfortunately, we also have not succeeded on finding that 10-chromatic /4,4,4,4,3j-
inflated C5 have thickness two. There were 5 edges that do not belong to any sub- graphs.
The /4,4,4,4,3j-inflated C5 graph has at most thickness three as shown in figure 6.7.
Figure 6.7: Decomposition of /4,4,4,4,3j-inflated C5
65


Tenth: Does the 10-chromatic 4-inflated C.7 have thickness two?
In order to answer the tenth question, we have applied the second technique. In this
technique we started with 4-inflated C7 on genus 0, which is 10-chromatic graph on 28
vertices and 154 edges, by (definition 9).
We also have not succeeded on finding that 10-chromatic 4-inflated C7 have thick- ness
two. There were 9 edges that do not belong to any subgraphs. The 4-inflated C7 graph has
at most thickness three as shown in figure 6.8.
Figure 6.8: Decomposition of 4-inflated C7
Eleventh: Is there a rectangle-visibility representation for K23 in 3- dimen-
sions?
In order to answer the eleventh question, we have applied the third technique. The search
for a representation of K22 ran on cluster computer for almost three months without
success. While a rectangle-visibility representation for K22 was found in approximately
one week.
In Appendix A, we summarized the best found results of applying first technique with
ACO, FA andSA algorithms in order to answer first, second and third questions.
66


6.2 Heuristic Methods Comparison
This study aimed at applying and comparing the different heuristic methods on solving
coloring problems in graphs and visibility representation for graphs in
3-dimensions. We have used four different heuristic methods throughout three
techniques (see chapter 5). The algorithms were evaluated using two criteria: the quality
of the solutions and the rate of time spending to find an optimal solution.
In first technique we implemented three different optimization algorithms (Ant Colony
Optimization (ACO), Firefly Algorithm (FA), and Simulated Annealing (SA)). We
have chosen ACO and FA due to the fact that different ants and different fireflies will
work almost independently, particularly that they are parallel implementation; while
SA has been chosen because it was the algorithm that used to find 9-critical graphs with
the same technique in [GS09], A further advantage of ACO and FA is that they are
even better than SA because ants and fireflies aggregate more closely around each
optimum solution without jumping around as in the case of SA. All these algorithms
produced almost the same quality of good solution. However, the difference is in the
amount of time that each algorithm spent to give the solution. We can see how the
swarm intelligence algorithms out- performed the simulated annealing in the following
figures for finding: orientable genus 0 9-critical thickness two graphs on 17 vertices whose
complements are K3- free (figure 6.9), orientable genus 0 9-critical thickness two graphs
on 25 vertices whose complements are AT4-free (figure 6.10), orientable genus 0 9-critical
thick- ness two graphs on 33 vertices whose complements are AT-free (figure 6.11), where x-
axis is the initial seeds andy-axis is the time on seconds.
Figure 6.9: First Technique Algorithms Performance (1)
67


9-CriticaI Thickness-two Graphs on 25 Vertices Whose
Complements are Iv4-free
14000
12000
10000
8000
6000
4000
2000
0
ACO
FA
SA
Figure 6.10: First Technique Algorithms Performance (2)
9-Critical Thickness-two Graphs on 33 Vertices Whose
Complements are Iv5-free
4500
4000
I ACO
I FA
ISA
10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Initial Seeds
Figure 6.11: First Technique Algorithms Performance (3)
Simulated annealing always accepts a selected better-cost local solution and it may also
accept a worse-cost local solution with a probability which is gradually decreased in
algorithms execution. As a result, it performs worse than swarm intelligence algorithms
since the last only accept the better-cost local solutions. Our results have shown that the
ACO is superior to FA in terms of time. Moreover, the results show that ACO
presented the best balance between the two criteria.
68


The third technique that we have implemented by the three different Swarm In-
telligence (SI) methods ( Ant Colony Optimization (ACO), Cuckoo Optimization
Algorithm (COA), and Firefly Algorithm (FA)). The main characteristic feature of the
SI algorithms is the fact that they simulate a parallel independent run strategy, where in
every iteration, a swarm of n (ants, cuckoos, and fireflies) have generated n solutions. Each
(ant, cuckoo, and firefly) works almost independently, and as a result the algorithm will
find the optimal solution very quickly. Figure 6.12 shows the performance of running
the algorithms in third technique to find visibility representation for complete graphs
Kn by rectangles in 3-dimensions.
Rectangle-Visibility Representation
12000
Figure 6.12: Third Technique Algorithms Performance
As before, the ACO outperformed FA, but COA is better than ACO. The major
disadvantage in the ACO is that while trying to solve the optimization problems, the
ants will walk through the path where the pheromone has been deposited, while the
local search has to performed much faster. Hence, local search will be performing at the
faster rate in COA than in the ACO, since the movement of cuckoo in COA is a
random walk. Moreover, the main advantage of COA is that it can solve optimization
problems based on that in every iteration, a fraction of worse solutions are abandoned
and in every generation cuckoos are moving to- wards an optimized solution by
replacing the solutions with the good ones. By the end of maximum iterations, we will
obtain the optimized solution.
Overall, the results indicate that all four methods can be used for solving com-
binatorial problems. The strength of the three SI algorithms is in parallel with the
search approach they employ. In computation terms, COA approach is faster,
69


although none of the SI algorithms take a long time like SA algorithm. The results
for SA suggest that while it is a valuable technique, it does not provide any
improvement over the other three techniques. The decision on whether to choose between
COA or ACO method is related to the evaluation criteria: the quality of the solutions
suggest both ACO and COA, while the fast rate time can be obtained by COA.
6.3 Future Work
The results of this thesis point to several interesting directions for future work: One
such direction would be to consider some combinations between ACO and COA
algorithms as an alternative approach to solve these and other problems even more
efficiently. A possible approach for this is to form a pipeline, passing the results from
one algorithm to another or in parallel, choosing the best result from the parallel runs
of several algorithms, or by having communication between the algorithms. Therefore,
we believe that hybrid or cooperative optimization strategies for combinatorial graph
theory problems are a promising future research.
Another possibility would be to design an exhaustive search for all decompositions of Ku
into two subgraphs which can be embedded on the double-torus, K20 into three subgraphs
which can be embedded on the double-torus, and K26 into four subgraphs which can be
embedded on the double-torus.
70


APPENDIX A
EXPERIMENT RESULTS
The following tables summarized the best found results of applying first technique with
ACO, FA and SA algorithms in order to answer first, second and third questions (see
research questions) by using CCM cluster computer. In Table Al, we showed the
results of the first technique by ACO algorithm. Table A. 2, shows the results of the
first technique by FA algorithm, while Table A.3, shows the results of the first
technique by SA algorithm.
Table A.l: The Results of First Technique by ACO Algorithm
Initial seed V X e Km-free Num. of Km start with Min Num. of Km
100 19 10 2 K, 23 5
2400 28 10 2 k4 265 71
60000 37 10 2 k5 312 43
700 21 11 2 k3 64 35
2000 31 11 2 K4 934 169
90000 41 11 2 k5 715 140
1100 23 12 2 k3 133 95
3000 34 12 2 K4 1262 69
12000 45 12 2 k5 901 162
800 33 17 3 k3 420 307
1600 49 17 3 k4 1343 332
80000 65 17 3 k5 1095 696
1400 35 18 3 k3 559 371
2000 52 18 3 k4 1506 606
19000 69 18 3 k5 1170 754
500 45 23 4 k3 790 199
3000 67 23 4 k4 1808 359
60000 89 23 4 k5 2200 459
100 47 24 4 k3 823 236
4000 70 24 4 k4 2010 352
19000 93 24 4 k5 3511 881
71


Table A.2: The Results of First Technique by FA Algorithm
Initial seed V X e A7m-free Num. of Km start with Min Num. of TS- J>^m
1300 19 10 2 K3 20 5
2200 28 10 2 k4 284 80
14000 37 10 2 k5 410 122
200 21 11 2 K3 69 38
7000 31 11 2 K4 945 106
50000 41 11 2 k5 765 144
500 23 12 2 k3 105 93
1000 34 12 2 K4 1227 115
61000 45 12 2 k5 886 162
200 33 17 3 k3 423 307
7000 49 17 3 k4 1360 335
31000 65 17 3 k5 1172 696
800 35 18 3 k3 580 371
5000 52 18 3 K4 1475 607
18000 69 18 3 k5 1222 755
1100 45 23 4 k3 799 198
3300 67 23 4 k4 1820 359
40000 89 23 4 k5 2311 460
1600 47 24 4 k3 816 236
6000 70 24 4 k4 1999 351
100000 93 24 4 k5 3312 880
72


Table A. 3: The Results of First Technique by SA Algorithm
Initial seed V X e A7m-free Num. of Km start with Min Num. of TS- J>^m
500 19 10 2 K3 16 5
2000 28 10 2 k4 299 71
31000 37 10 2 k5 50 45
1400 21 11 2 K3 70 37
5000 31 11 2 K4 832 109
100000 41 11 2 k5 690 142
800 23 12 2 k3 126 97
2300 34 12 2 K4 1409 229
70000 45 12 2 k5 891 162
1400 33 17 3 k3 441 307
2500 49 17 3 k4 1402 390
30000 65 17 3 k5 1210 699
900 35 18 3 k3 512 371
2800 52 18 3 K4 1380 606
30000 69 18 3 k5 1078 771
500 45 23 4 k3 654 198
1500 67 23 4 k4 1795 359
70000 89 23 4 k5 1982 460
1600 47 24 4 k3 791 237
4000 70 24 4 k4 1989 351
20000 93 24 4 k5 2155 883
73


APPENDIX B
NEW NINE-CRITICAL GRAPHS
We will introduce thirty 9-critical thickness two graphs on 17 vertices with W3- free
complements, thirty 9-critical thickness two graphs on 25 vertices with W4-free
complements and thirty 9-critical thickness two graphs on 33 vertices with AT-free
complements by using (ACO, FA, and SA) algorithms. The graphs were chosen for the
uniqueness of degree sequences. To that end, each graph G is identified by its degree
sequence.
In order to display a thickness two embedding for each graph we used Groups and Graphs
[Mckl2], The final graphics were generated by Mathematica [Woll2], The 90 new 9-
critical thickness two graphs follow.
B.l Thickness two graphs on 17 vertices whose complements are A^-free
B.1.1 Ant Colony Algorithm (ACO)
I Jcgree Sequence
[13,13, 12. II. 10,10,10, ID, 10, 10. 10, 10, 9, 9, 9, 9. 9
74


Degree Sequence
{13,12, 12, 12,12, 11, 11, 10, 10, 10, 10, 10, 9, 9, 9, 9, 9}
75


Degree Sequence
{13, 12, 11,11, 11, 11, 11, 10, 10, 10, 10, 10,10, 9, 9, 9, 9}
76


Degree Sequence
{12, 12, 11,11,11,11, 11,10, 10, 10, 10,10, 9, 9, 9, 9, 9}
77


Degree Sequence
{13,12, 11, 11,10, 10, 10, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9}
78


Degree Sequence
{12,12, 11, 11, 10, 10, 10, 10,10, 9, 9, 9, 9, 9, 9, 9,9}
80


Degree Sequence
{12, 12, 12, 11,10, 10, 10, 10, 10, 10, 10, 10, 9,9, 9, 9,9}
81


Degree Sequence
{13,11,11,11,11, 11,10, 10, 10, 10,10, 9, 9, 9, 9, 9, 9}
82


Degree Sequence
{13,13,13,11, 11,10,10,10, 10,10, 9,9, 9,9,9, 9,9}
83


B.1.3 Simulated Annealing Algorithm (SA)
84


Degree Sequence
{13,12, 11,11, 11,11, 11, 10, 10, 10,10, 9, 9, 9, 9, 9, 9}
Degree Sequence
(12, 12,12, 11, 11,11,10, 10, 10, 10,10, 10,9,9,9,9,9}
85


Degree Sequence
{12,12,12,11,11,11,10, 10, 10,10,9,9,9,9,9,9,9}
86


Degree Sequence
{13, 12, 11, 11,11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 9, 9, 9}
87


Degree Sequence
{12,11,11,11,11,11,10, 10,10,10, 10,10, 9, 9, 9, 9,9}
88


Degree Sequence
{12,12, 12, 11,11,10, 10, 10,10,10, 10, 9, 9, 9, 9, 9, 9}
89


Full Text

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HEURISTIC METHODS A PP L I ED TO DI FF I CU L T GRAPH THEO R Y P R OB LE MS b y ROQYAH ALALQAM B.S., King Saud Uni v ersi t y 2007 A t h esi s s ubm i tt ed t o t h e F a cu l t y of t h e Graduate S c h o o l of t h e Un i v er si t y of Colorado in partial fu l fi ll m e n t of t h e r eq u i re m e n t s for th e degree of Master of Science in Com pu t er S ci e n ce 2012

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ii Th i s t h es i s for t h e Master of Science d eg r ee b y R o q y ah Al a l q am has b een appr o v ed for t h e Master of Science in Com pu t er Science b y Dr. Ellen Ge th n e r, Ad v i so r Dr. Bogdan Ch l eb u s Dr. Mi c hael F er r ar a November 5, 2012

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iv Alalqam, R o q y ah, R. (Master of Science in C om pu t er S ci en c e) Heuristic Me th o d s Applied t o D i ffi cu l t Graph Theory Pr o b l em s Th es i s d i r ect ed b y Ass o ci a t e Pr of esso r Ellen Ge th n e r A BST R A C T Th e pur p ose of t h i s study i s t o offer new i n si g h t and t o o l s t o w a r d th e pu r su i t of t h e l ar g est c h r om a t i c n u m b er in t h e class of t h i c k n ess t graphs. Th e t h i c k n es s of a graph i s th e mini m um n u m b er t of planar subgraphs where t h e graph can b e decom p osed. Determining t h e t h i c k n ess of a gi v en graph i s kn o wn t o b e an NP co mp l et e p r ob l e m. Th e c hromatic n u m b er i s t h e sm a ll est n u m b er k su c h t h a t t h e v er t i ces of a graph can b e pro p erly co l or ed wi t h k colors. S i m il a r l y determining t h e c hromatic n u m b er of a gi v en graph i s kn o wn t o b e an NP com p l e te problem. An o th e r aim w as t o i n v est i ga t e th e l a r ge st cl i q u e in th e class of 3 d i m e n si on a l vi s ibili t y re p re se n t at i o n of graphs in whi c h v e rt i ce s are map p ed t o rectangles fl oa t i n g in R 3 parallel t o th e x, y a x i s, wi t h edges r ep r ese n t ed b y v e rt i ca l li n es of si g h t In t h i s t h e si s, heuristic sear c h meth o d s are c o m b i n e d wi t h graph t h eo re ti c meth o d s t o d i sc o v er more a b o u t t h e up p er b ound of t h e c h r om a t i c n u m b er for t h e Ea r th M o on p r ob l e m and i t s gen e ra li za t i on t o higher th i c k n ess and d i ffer e n t s u rf ace s. Ad d i t i on a ll y t h ey a r e used t o ex p l or e more a b o u t t h e up p er b ound of t h e largest size of th e cl i qu e t h a t has a re ct an g l e vi s i b ili t y r ep r ese n t a t i on in 3 d i m e n si o n s. By i mp l em e n t i n g h eu r i s ti c sea r c h m e th o d s in Py t h o n and C l an g u ag es on t h e s ph er e w e found new 90 9 c hrom a ti c t h i c k n es s t w o graphs whose c om p l em e n ts a r e K m free. Also b y applying th e same m e th o d s on th e t oru s w e found 1 9 c h r om a t i c t h i c k n ess t h re e graph, and 24 c h r o m at i c th i c kn e ss four graph. On t h e double t oru s w e found 14 c hrom a ti c t h i c k n es s t w o graph, 20 c hrom a ti c t h i c k n es s th r ee graph, and 26 c h r o m at i c t h i c k n es s four graph. On th e other hand, for t h e vi s i b ili t y r ep r ese n t a t i on b y r ec ta n g l es in 3 d i m e n si on s w e h av e found a r ep r ese n t a t i on of K 2 2 F u r t h er m o re v a r i ou s h eu r i s ti c sear c h t e c hn i q u es a r e compared and d e ta il ed d i scu s si on of t h e heuristic sea r c h al go r i th m s and t h ei r i mp l em e n t at i o n i s pr o vided. Th e form and co n t e n t of t h i s abstract a r e ap p r o v e d. I recommend i t s pub li ca ti o n. Appr o v ed: Dr. Ellen Gethner

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iv DEDIC A TION I d ed i ca te t h i s t h es i s t o m y w onderful famil y P a r ti c u l ar l y t o m y p a r e n t s, under st a nd i n g and p a t i e n t h usband, Mahmoud, and t o our go r ge ou s son Amjed, who i s t h e j o y of our li v es.

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v ii A C K N O WL E DGE MEN TS I am d eep l y g ra t efu l t o all t h o se who willingly h e l p ed me acco mp li sh t h i s stud y I w ould li k e t o ex p r ess m y si n ce r e t h a n ks and g r at i t ud e t o m y advisor, Pr o fess or Ellen Gethner, for her gu i d a n ce, su p p or t m o t i v at i o n and enco u r a gem e n t t h r ou g h o u t t h e p eri o d t h i s w ork w as carried ou t Her re ad i n e ss for co n su l t at i o n a t a ll t i m es, her ed u c at i v e comme n ts, her a d v i ce during th e wr i t i n g of t h i s t h esi s, and her concern and ass i st an c e e v en wi t h p r ac ti c al t h i n g s h a v e b een i n v aluable. I w ould li k e t o ex p r ess m y gr a t i tu d e t o Dr. Thom S u l a n k e for h i s v ery u sefu l com m e n t s and sug ge st i on s Hi s sincere i n t er es t in t h i s r es ear c h t h em e i s gr ea t l y ap p r eci a t ed I w ould li k e t o thank t h e ce n t er for co mpu t at i o n a l m a t h em a ti c s ( CCM) for l et t i n g me used th e cluster c om pu t er t o do m y co mpu t at i o n s. I w ould also li k e t o t h an k all t h e t u to r s of t h e D ep ar t m e n t of Com pu t er Science & En g i n eer i n g at t h e Un i v er si t y of Colorado a t De n v er, who t au g h t me during m y stud y Th e staff a t t h e D epa r t m e n t for t h ei r friendly c o o p eration.

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v ii T ABLE OF C O NT EN TS Ch a p t er 1. I n tr o duct io n . . . . . . . . . . . . . . . . . 1 1.1 S ta t em e n t of t h e Pr ob l e m s . . . . . . . . . . . . 3 1.2 Pur p ose of t h e S t ud y . . . . . . . . . . . . . 4 1.3 Resear c h Qu est i o n s . . . . . . . . . . . . . . 4 1.4 S i gn i fi ca n ce of t h e S t ud y . . . . . . . . . . . . 5 1.5 Li m i t a t i on s of t h e S t ud y . . . . . . . . . . . . 5 2. Ba c kground of the Study . . . . . . . . . . . . . . . 6 2.1 W h y Graphs? . . . . . . . . . . . . . . . 6 2.2 Basic S t r u ct u re s and D efi n i t i o n s . . . . . . . . . . . 6 2.2.1 Gr a ph s . . . . . . . . . . . . . . . . 7 2.2.2 Common F a m ili es of G raph s . . . . . . . . . . 8 2.2.3 Graph O p e ra t i on s . . . . . . . . . . . . 9 2.2.4 Planar G raph s . . . . . . . . . . . . . 10 2.2.5 r I n fl at ed G raph s . . . . . . . . . . . . 12 2.3 Th i ckness of a Graph . . . . . . . . . . . . . . 13 2.4 Grap h Coloring . . . . . . . . . . . . . . . 14 2.5 Vi si b ility R ep r ese n ta t i on . . . . . . . . . . . . . 17 2.5.1 Ba r Vi si b ili t y Gr a ph s . . . . . . . . . . . 17 2.5.2 R ect an g l e Vi si b ili t y Gr ap h s . . . . . . . . . . 18 3. Review of Related Literature . . . . . . . . . . . . 19 3.1 Graph Coloring Problem . . . . . . . . . . . . 19 3.1.1 I n t r o du c ti o n . . . . . . . . . . . . . 19 3.1.2 Related S t ud i es . . . . . . . . . . . . . 20 3.1.3 Conclusion . . . . . . . . . . . . . . 23

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v iii 3.2 Vi si b ili t y R ep r ese n ta t i on Problem . . . . . . . . . . 23 3.2.1 I n t r o du c ti o n . . . . . . . . . . . . . 23 3.2.2 Related S t ud i es . . . . . . . . . . . . . 24 3.2.3 Conclusion . . . . . . . . . . . . . . 29 4. Optimization Sear c h Meth o ds . . . . . . . . . . . . 30 4.1 S w arm I n t el li gen ce Al go r i th m s (SI) . . . . . . . . . . 30 4.1.1 A n t Colo n y Op ti m i zat i o n Al go r i t hm ( A CO) . . . . . 31 4.1.2 Cu ck o o Op t i m i za t i on Al g or i t hm (C O A) . . . . . 35 4.1.3 Firefly Al go r i t hm ( F A) . . . . . . . . . . . 38 4.2 S i m u lated An n ea li n g Algorithm ( S A) . . . . . . . . . 41 5. Resear c h Meth o dology . . . . . . . . . . . . . . 45 5.1 F i rs t T e c hn i q u e . . . . . . . . . . . . . . . 45 5.2 Second T e c hn i q u e . . . . . . . . . . . . . . 47 5.3 Third T e c hn i q u e . . . . . . . . . . . . . . 48 6. Findings and Discussions . . . . . . . . . . . . . . 50 6.1 Res u l ts of t h e S t ud y . . . . . . . . . . . . . . 50 6.2 Heuristic Met h o d s Co mpar i so n . . . . . . . . . . . 67 6.3 F u tu r e W ork . . . . . . . . . . . . . . . . 70 Ap p endix A. E xperiment R esu l ts . . . . . . . . . . . . . 71 B. N ew N i n e C r i ti ca l G ra ph s . . . . . . . . . . . . . 74 Bibliograp h y . . . . . . . . . . . . . . . . . . 121

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ix Figure L I ST OF F I GU RES 1.1 Rep r es e n t i n g K 6 b y r ect an g l es in 3 d i m en si o n s . . . . . . . 2 2.1 S i mp l e Graph G . . . . . . . . . . . . . . . 7 2.2 Graph G and Subgraph H . . . . . . . . . . . . 8 2.3 Ex am p l es of C om p l et e G r ap h s . . . . . . . . . . 9 2.4 Com p l e te Bi p a rt i t e Graph K 2 3 . . . . . . . . . . 9 2.5 A K 4 Planar Graph . . . . . . . . . . . . . . 10 2.6 Ku r at o wsk s Graph K 5 and K 3 3 . . . . . . . . . 12 2.7 Ex am p l e of r i n fl a t i on G r ap h s . . . . . . . . . . . 13 2.8 De com p o si t i o n of K 9 . . . . . . . . . . . . . 14 2.9 F our colored map . . . . . . . . . . . . . . 15 2.10 8 Ch r o m at i c C om p l et e Graph K 8 . . . . . . . . . 16 2.11 1 Ch r o m at i c E mp t y Graph . . . . . . . . . . . . 16 2.12 2 Ch r o m at i c B i par t i t e Graph K 3 4 . . . . . . . . . 16 2.13 B ar Vi s i b ili t y Graph . . . . . . . . . . . . . 17 2.14 R ect a ngl es Vi si b ili t y Graph . . . . . . . . . . . 18 3.1 Or i e n t a b l e S u r fac es of ge n us g = 0, 1, and 2 . . . . . . . . 19 3.2 Bi e m b ed d i n g of K 13 . . . . . . . . . . . . . 23 3.3 K 8 has a r ec ta n g l e v i si b ili t y graph . . . . . . . . . 25 3.4 Rep r es e n t i n g K 9 b y K 4 K 5 b l o c k s . . . . . . . . . 26 3.5 Rep r es e n t at i o n of K 20 . . . . . . . . . . . . 26 3.6 Rep r es e n t i n g K 7 b y un i t squares . . . . . . . . . . 27 3.7 A n y K m,n has a r ec ta n g l e v i si b ili t y r ep r ese n t a t i on . . . . . 27 3.8 A n y K n has a d i s cs v i si b ili t y r ep r ese n ta t i on . . . . . . 28 4.1 A n ts find sh or t est p a t h b e tw een t h ei r nest and f o o d s ou r ces . . . . 32 4.2 Pseudo c o de of A CO a l g or i t hm . . . . . . . . . . 33

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x i 4.3 Ex am p l e for Applying A CO Al g or i t hm . . . . . . . . 33 4.4 T race for Applying A CO Al go ri t hm . . . . . . . . 35 4.5 Pseudo c o de of CS al go ri t hm . . . . . . . . . . . 36 4.6 Ex am p l e for Applying C O A Al g or i t hm . . . . . . . . 37 4.7 T race for Applying C O A Al go ri t hm . . . . . . . . 38 4.8 Pseudo c o de of F A al g or i t hm . . . . . . . . . . 39 4.9 Ex am p l e for Applying F A Al go ri t hm s . . . . . . . . 40 4.10 T race for Applying F A Al g or i t hm s . . . . . . . . . 41 4.11 Pseudo c o de of SA a l go r i t hm . . . . . . . . . . 42 4.12 Ex a mp l e for Applying SA Al g or i t hm s . . . . . . . . 43 4.13 T race for Applying SA Al g or i t hm s . . . . . . . . . 44 6.1 New 9 Cr i t i ca l Graph whose Com p l e m e n t i s K 3 f re e . . . . 52 6.2 New 9 Cr i t i ca l Graph whose com p l e m e n t i s K 4 f re e . . . . 53 6.3 New 9 Cr i t i ca l Graph whose com p l e m e n t i s K 5 f re e . . . . 53 6.4 De com p o si t i o n of K 19 . . . . . . . . . . . . . 56 6.5 De com p o si t i o n of K 25 . . . . . . . . . . . . . 57 6.6 De com p o si t i o n of K 24 . . . . . . . . . . . . . 58 6.7 De com p o si t i o n of { 4 4, 4 4, 3 } i n fl a t ed C 5 . . . . . . . 65 6.8 De com p o si t i o n of 4 i n fl a t ed C 7 . . . . . . . . . . 66 6.9 F i rs t T e c hn i q u e Al g or i t hm s P erformance (1) . . . . . . 67 6.10 First T e c hn i qu e Al go r i th m s P erformance (2) . . . . . . 68 6.11 First T e c hn i qu e Al go r i th m s P erformance (3) . . . . . . 68 6.12 Third T e c hn i q u e Al go r i t hm s P erformance . . . . . . . 69 B.1.1 Ant Colony Algorithm (ACO) . . . . . . . . . . . 74 B.1.2 Firefly Algorithm (FA) . . . . . . . . . . . . . 79 B.1.3 Simulated Annealing Algorithm (SA) . . . . . . . . . 84 B.2.1 Ant Colony Algorithm (ACO) . . . . . . . . . . . 90 B.2.2 Firefly Algorithm (FA) . . . . . . . . . . . . . 95

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x i B.2.3 Simulated Annealing Algorithm (SA) . . . . . . . . 100 B.3.1 Ant Colony Algorithm (ACO) . . . . . . . . . . . 105 B.3.2 Firefly Algorithm (FA) . . . . . . . . . . . . . 110 B.3.3 Simulated Annealing Algorithm (SA) . . . . . . . . 1 15

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xii L I ST OF T AB L ES Table 1 Th e f un ct i on s H ( g ) for a su r fa ce of ge n us g . . . . . . . 21 2 Th e f un ct i on s B ( g ) for a su r fa ce of ge n us g . . . . . . . 22 3 L o w er and up p er b o und s for re ct an g l e v i s i b ili t y r ep r ese n ta t i on s . . 28

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1 1. I n t r o ducti on A graph i s an abstract r ep r ese n t a ti o n of r el a ti o n sh i p s. It i s defined as a diagram co n si st i n g of p o i n t s, called v e rt i c es, some of whi c h are connected b y li n es, called edg es. F or ex a mp l e, a ssi g n i n g r eg i st er s t o t em p or a r y v ar i a b l es can b e co n v er t ed i n t o an eq u i v al e n t graph b y l e tt i n g ea c h v a r i ab l e t o b e a v er t ex and co nn ect i n g t w o v er t i ces b y an edge if t h e co r re s p on d i n g v ar i a b l es are i n t er fe re Sup p ose w e a r e as k ed t o help o u t a com p il e r t o r ed u c e th e n u m b e r of r eg i st er s needed in su c h a w a y t h at i f t h er e i s an edge b e t w een t w o v ar i a b l es, t h en w e sho u l d not assign t h e same r eg i st er t o t h em since t h i s r egi s te r needs t o hold th e v a l u es of b oth v a r i ab l es a t one p o i n t of t i m e. Th e i d ea i s to use t h e fe w est p os si b l e n u m b er of r eg i st er s. Can this b e done? Here, w e b ri e fl y i n t r o du c e t h e graph c ol o r i n g p r ob l e m, a classic p r o b l em in graph t h eo r y Th e graph coloring p r ob l e m i s t o assign co l or s t o t h e v er t i ces of a graph in su c h a w a y t h at no t w o v e rt i ce s connected b y an edge share t h e same co l or, and t h e aim i s t o use th e fe w est p oss i b l e n u m b er of colors. An o th e r ex a mp l e, th e ci r cu i t b oard can b e co n v erted i n t o an e qu i v a l e n t graph b y l et t i n g th e co m p on e n t s t o b e v er t i ces and t h e wires t h a t connected cer t a i n pai r s of t h e c om p o n e n t s t o b e edges. Sup p ose w e a r e as k ed t o help ou t a c i rcu i t d esi g n er t o m o del t h e c i rcu i t b oard in su c h a w a y t h a t con n e ct i n g w i re s li e v er t i ca ll y or h o ri z o n t a ll y and do not cross and t h at t h e c om p o n e n t s li e p e r p e nd i cu l ar t o t h e wires. Can this b e done? Th e t y p e of graph t h a t i s most li k e l y t o h e l p t h e ci r cu i t designer i s a r ect a ngl e v i si b ili t y graph. F rom t h i s p oi n t w e i n tr o du ce t h e v i si b ili t y r ep r ese n ta t i on prob lem for graph. Th e v i si b ili t y r ep r ese n ta t i on p r ob l e m i s t o det er m i n e whether a graph i s r ep r ese n ta b l e or not. Th e graph coloring problem has an es se n t i al r o l e in co mpu t er science. It m o d el s m a jor r ea l w o r l d problems. T h e m a jor ap p li ca t i on a r eas are: t i m e t a b li n g and s c h e du li n g, frequency ass i gn m e n t r egi s te r all o cation, and p r i n t ed ci r cu i t t es ti n g On t h e o th e r hand, t h e v i si b ili t y r ep r ese n t a ti o n p r o b l em has a l so l ar g e n u m b er of a pp li cat i o n s re l at ed to com pu t er science su c h as VL S I design, CASE t o o l s, c i rcu i t b oard l a y o u t and a n i m a t i on problem. In th i s t h esi s, w e study a gen er a li za ti o n of a p r o b l em p osed b y Gerhard R i n ge l i n

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2 1959 [Rin59], whi c h i s a n a t ura l gen er a li za ti o n of t h e F our Color problem. It i s a long standing o p en problem, whi c h i s d e te rm i n i n g t h e l a r ge st c hrom a ti c n u m b e r of an arb i t r ar y th i c kn e ss t graph for a n y t 2 and a n y ge n us g In ad d i t i o n, w e s tu d y 3 d i m en s i on s v i si b ili t y t h at r ep r ese n t v er t i ces b y 2 d i m e n si on a l rectan g l es p l aced in p l a n es parallel t o t h e x y plane. T w o v er t i ces are connected b y an edge if and only if t h ey can see ea c h other in t h e direction t h a t i s p er p endicular t o t h e z axis. Th i s t y p e of r ep r ese n t a ti o n w as i n t r o du ced as a ge n er al i za t i on of t h e 2 d i m en si o nal v i si b ili t y r ep r es e n t at i o n [ B EF + 94 ]. T h e 3 d i m e n si o nal rectan g l e v i si b ili t y r ep r es e n t at i o n r ecei v ed a wi d e at t e n t i o n. H o w e v er, w e f o cus on t h e maxi m um size of a co mp l et e graph wi t h a 3 d i m en si o n s v i si b ili t y r ep r ese n ta t i on b y rectangles. See fi g u r e 1.1 for an e xa mp l e for v i si b ili t y r ep r es e n t at i o n of co mp l et e graph K 6 b y r ect an g l es in 3 d i m e n si on s Figure 1.1: Re p re se n t i n g K 6 b y r ect an g l es in 3 d i m en si o n s Th i s t h esi s p r es e n t s th e r es u l ts of using heuristic sea r c h meth o d s i mp l em e n ted i n Py t h on and C l a n gu a ge s t o find k c hrom a ti c t h i c k n ess t graphs on n v er t i ces and ge n us g of o r i e n t ab l e su r fa ce, and t o find t h e r ep r ese n t a t i on of l a r ges t co mp l et e graph K n b y rectangles in 3 d i m e n si on s Ad d i t i on a ll y a v arie t y of d i ffe re n t heuris t i c sear c h m e th o d s a r e compared and d e ta il ed d i s cu ssi o n of t h e heuristic sea r c h a l go r i th m s i s pr o vided. Th e st r u ct u r e of t h i s thesis i s go i n g t o b e as foll o ws: Ch a p t er 2 p r e se n t s th e rele v a n t ba c kground i n fo r m a ti o n in t h e foll o wing o rd e r: w h y graphs?, b as i c st ru c tu r es and d efi n i t i on s t h i c k n es s of graphs, graph coloring, and v i si b ili t y r ep r ese n t a ti o n for graphs. Ch a p t er 3 i s a review li t er a t u r e of related st ud i es.

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3 In Ch ap t er 4, a p r ese n t a t i on of th e d i ff er e n t h eu ri s ti c meth o d s will b e p r ese n t ed In Ch ap t er 5, w e are finally m o ving i n to t h e main p a r t of t h i s t h es i s, namely t h e m e th o d o l og y of t h e stud y i mp l e m e n t at i o n and a d es cr i p ti o n of t h e c o de used t o acco mp li sh t h i s stud y Th e cl os i n g c h ap t er Ch ap t er 6, will co n t a i n th e re su l t s of t h i s study and a d i sc u ssi o n of t h e r esu l t s. Ad d i t i o nal l y a co mp a ri s on of t h e d i ffer e n t heuris t i c meth o d s for t w o d i ffe re n t p r o b l em s and sugg est i o n s for fu t u r e w ork will b e pr o vided. Th er e will b e an Ap p endix A t h at s ummar i zes t h e r esu l t s of t h e ex p eri m e n t and Ap p endix B co n ta i n s 90 9 cr i t i cal graphs wi t h t h i c k n ess t w o and ge n u s 0. 1.1 Sta te me n t of t he P roblem s Let G ) to b e t h e th i c kn e ss of graph G A graph G has thic k ness t wi t h res p ect t o th e ge n us g if G can b e decom p osed i n to t subgraphs and no fe w er t h an t Th i s p a rt i t i o n ma k es t co p i es of t h e v er t i ces of G and ea c h edge of G i s assigned t o one of t h e t co p i es. More o v er, ea c h subgraph of G sho u l d b e g e m b ed d a b l e. Let G t,g d eno t es an o r i e n t ab l e graph wi t h t h i c k n ess t and ge n us g Let G b e a graph; t h e r i n fl a t i on of G i s t h e l e xi c og r ap h i c pr o duct G [ K r ], and i s denoted b y G [ r ]. Th e c hromatic n u m b er of a graph G denoted ( G ), i s t h e mini m um n u m b er of co l or s needed t o color t h e v er ti c es of G su c h t h at no t w o a d ja ce n t v er ti c es recei v e th e same color. Th e qu e st i on w e w a n t t o study is: What is the b est up p er b ound for the ch r omatic num b er of any genus g thickness t orientable g r aph? Vi si b ili t y r ep r ese n ta t i on s of graphs map v e rt i ce s t o sets in Eu cl i d e an space and ex p r ess edges as v i s i b ili t y r el at i o n s b e tw een t h ese sets. F or graph G = ( V E ) th e 3 d i m en si o n a l v i si b ili t y r ep r es e n t at i o n b y r ec ta n g l es i s defined as an ar r a n gem e n t of d i s joi n t r ect an g l es in R 3 su c h t h at t h e p l a n es det er m i n ed b y t h e rectangles a r e v er t i cal t o t h e z a xi s and t h e sides of th e r ec ta n g l es are parallel t o t h e x a xe s or y axes. A gi v en graph G i s said t o b e r ep r es e n t ab l e if and only if i t s n v er t i ces can b e ass o ciated wi t h n d i sjo i n t r ec ta n g l es parallel t o t h e x and y axes in R 3 su c h t h a t v e rt ex v i and v j are ad ja ce n t in G if and only if t h ei r co r re s p on d i n g rectangles R i and R j are z visible. Th e q u est i on w e w a n t t o study is: What is the up p er b ound on the size of the la r gest clique that c an b e r ep r esent e d in 3 dimensions? Th e heuristic sear c h meth o d s are applied t o find t h e up p er b ound of t h e c hro m a ti c n u m b er of a n y ge n us g t h i c k n ess t o ri e n ta b l e graph. In a dd i ti o n, th e y a r e a pp li ed t o find th e maxi m um s i ze of a co mp l et e graph wi t h 3 d i m e n si on s v i si b ili t y r ep r ese n t a t i on b y r ect a ngl e s.

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4 1.2 Pur p ose of t he St udy Th er e are ma n y i n t er est i n g unsolved coloring p r ob l e m s for graphs t h a t can b e decom p osed i n t o m u l t i p l e planar l a y e rs Th e decision p r ob l e m t h i c k n ess, whi c h det er m i n es for an i npu t graph G and i n t eg er t if G ) t i s NP Hard [ Mi t 81 ]. F u r t h er mor e, t h e decision problem k c ol o r ab l e graph, whi c h det er m i n es if th e re i s a mapping of k co l or s t o v e rt i ce s su c h t h a t all ad j ace n t v er ti c es h av e d i ff er e n t co l or s, i s NP Hard [GJ79]. Th e co m b i n at i o n of t h ese t w o NP Hard p r o b l em s m a k e s so l v i n g m u l t i th i c k n ess graph coloring prob l em s v e r y c hal l e ngin g T h us, it w ould b e d es i rab l e t o h a v e a collection of graph fa m ili es for whi c h b o t h t h e t h i c k n ess and c h r om a t i c n u m b e r are und e rs t o o d su c h as com p l e te graphs K n In t h i s t h es i s, heuristic sear c h meth o d s a r e co m b i n ed wi t h graph t h eo r et i c m e th o d s to gen er a t e graphs wi t h s p ecific c hromatic n u m b ers or th i c kn e ss. Th e goal of cr ea ti n g and exploring t h ese graphs i s t o d i sc o v er more a b ou t t h e b ou nd s of t h e c hromatic n u m b er for t h e Ear t h M o o n problem and i t s ge n er al i za t i o n t o higher th i c kn e ss and d i ffer e n t su r fa ces. F u r t h er mor e, se v eral data p r ese n t a ti o n p r ob l em s i n v o l v e dr a wing g r ap h s Th e study of t h i s dr a wing w as originally m o t i v at ed b y VL S I l a y ou t A si gn i fi c a n t p r ob l e m i s t h e one of determining v i si b ili t i es b e tw een d i ffer e n t e l ect r i ca l com p o n e n t s. S i n c e VL S I st ru c tu r es are laid in a p l a n e, and th e com p one n ts are b ounded b y i so th e ti c or i e n t ed rectangles, t h en t h e v i si b ili t i es can b e st ud i ed wi t h i n t h e t w o fa m ili es of parallel s i d es, i nd e p e nd e n t l y [LMW87]. Th e pur p ose of t h e study i s t o r ep r ese n t t h e p r ob l em of seg m e n t v i si b ili t y in graph form and t o d i sc o v er more a b ou t t h e b ou nd s of t h e l ar g est size of co mp l et e graphs t h at h a v e r ec ta n g l es visi bili t y r ep r ese n t a t i on in 3 d i m en s i on s b y co m b i n i n g heuristic sear c h meth o d s wi t h graph t h eo re ti c meth o d s 1.3 Resear c h Questions Th e m a jor qu e st i on s th a t are a dd r esse d b y t h i s s tu d y a r e t h e foll o wing: 1. D o es t h er e exist a graph G 2 0 wi t h ( G 2 0 ) = 10, 11, or 12? 2. D o es t h er e exist a graph G 3 0 wi t h ( G 3 0 ) = 17 or 18 ? 3. D o es t h er e exist a graph G 4 0 wi t h ( G 4 0 ) = 23 or 24 ? 4. D o es t h er e exist a graph G 3 1 wi t h ( G 3 1 ) = 19? 5. D o es t h er e exist a graph G 4 1 wi t h ( G 4 1 ) = 25 ? 6. D o es t h er e exist a graph G 2 2 wi t h ( G 2 2 ) = 14 ?

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5 7. D o es t h er e exist a graph G 3 2 wi t h ( G 3 2 ) = 20? 8. D o es t h er e exist a graph G 4 2 wi t h ( G 4 2 ) = 26? 9. D o es t h e 10 c hrom a t i c { 4 4, 4 4, 3 } i n fl a t ed C 5 h a v e th i c kn e ss t w o ? 10. Dose t h e 10 c h r o m at i c 4 i n fl a t ed C 7 h a v e th i c kn e ss t w o ? 11. Is t h er e a re ct an g l e vi s i b ili t y r ep r ese n ta t i on for K 23 in 3 d i m en si o n s? 1.4 Significance of t he St udy Ex cep t for t h e t or o i dal graphs whose su r fac e has Euler c h a ra ct er i st i c 0 [BM92], t h e Ea r t h M o o n problem and its ge n er al i z at i o n t o higher t h i c k n es s and d i ffer e n t su r fa ces it has n o t b een so l v e d y et. So, w e a tt em p t ed to r ea c h new re su l t s and so l v e t h e p r ob l em b y t h i s s tu d y F u r t h er mor e, th e cu r re n t b est up p er b ound on t h e size of th e largest co mp l et e graph K n t h at can b e re p re se n t ed b y rectangles i s 50 [ BD HS 97 ][ S 09 ], and t h e l o w er b ound i s 22 [ BEL + 93 ]. Th r o u gh o u t t h i s study w e a tt em p t ed to narr o w t h e gap b e tw een t h e kn o wn up p er and l o w er b ound for r ep r ese n ta t i on of K n b y r ect a ngl es 1.5 Li mi ta t io ns of the Study Finding t h e l a rg est c h r om a t i c n u m b er of a n y ge n us g t h i c k n ess t o ri e n ta b l e graph i s co mp li ca t ed for t h e r e aso n t h at for gi v en an a rb i t r a ry graph G and a fixed p os itive i n t ege r t 2, and ge n us g 0, v e ri f yi n g t h at G ) = t i s an NP c om p l et e p r ob l e m. S i m il a r l y gi v en an a r b i t r ar y graph G and fixed p o si t i v e i n t eg er k > 2, v erifying t h a t ( G ) = k i s also NP com p l e te. In t h at case, t h e approa c h of st ar t i n g wi t h a graph of kn o wn t h i c k n es s and finding i t s c h r om a t i c n u m b er, or v i ce v er sa will n o t often end in a c h i e v em e n t [ABG10]. S i n ce bar v i si b ili t y graphs are n a t u r al l y planar [ Wi s 85 ] [TT86], and r ec ta n g l e v i si b ili t y graph i s t h e union of t w o bar v i si b ili t y graphs, t h en re ct an g l e v i si b ili t y graph naturally has t h i c k n es s at most t w o, seen b y p a rt i t i o n i n g t h e edges i n t o t w o sets co rr es p on d i n g t o v er ti c al and horizo n tal v i si b ili t i es It w ould b e u sef u l t o h a v e a si mp l e c har a ct er i za ti o n of r ec t an g l e v i si b ili t y graphs, bu t no c har a ct er i zat i o n h a s y et b een found; n ei t h er has th e p r ob l e m b een sh o wn to b e NP co mp l et e, t h ou g h it i s an NP co mp l et e p r o b l em t o r ec og n i ze t h i c k n ess t w o graphs [DH97].

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6 2. Ba c kground of the St udy Th i s c h ap t er g i v es a brief i n t r o du ct i on t o t h e t er m i nol og y used l a t er in t h i s t h esi s. Th e aim i s t o p r o v i d e t h e reader wi t h t h e minimal necessary graph t h eo r et i c ba c kground to understand t h e co n cep t s and re su l t s of th i s d i sser t at i o n. 2.1 W h y G ra phs ? F und am e n t a ll y co mpu t er science i s a science of a b st r act i o n as Aho and Ho p croft illustrated [AH74]. Therefore, computer sci e n ti s ts m ust create a b st r act i o n s of r ea l w o r l d p r o b l em s t h a t can b e r ep r ese n t ed and m an i pu l a t ed in a co mpu t er F or ex a mp l e, for su cce ssfu l s c h ed u li n g of final exams, w e h a v e t o con s i d er th e ass o cia t i on s b e tw een co u r ses, st ud e n t s and r o oms. Su c h set of con n e ct i on s b e t w een i t em s i s m o d el e d b y graphs. A graph i s com p osed b y some e l em e n t s ca ll ed v er ti c es, and t h e r el a ti o n s among t h em a r e ca ll ed edg es. Th e b as i c i d e a of graphs w as i n t r o du c ed in 1736 b y t h e g r ea t m a th e m at i ci a n Leonhard Euler. He used graphs t o so l v e t h e famous K ¨ o n i gs b e rg Br i d g e problem. Euler studied t h e p r ob l e m of K ¨ o n i gs b e rg Br i d ge and constructed a structure t o so l v e t h e problem whi c h i s called Eulerian graph [SHL07]. In 1840, A.F M ob i u s g a v e th e i d ea of Co mp l et e Graph and B i par t i t e Graph. Euler and Ku r a t o ws ki for m ulated t h ei r famous c h a r ac te r i zat i o n of Planar Graphs. Euler m e n t i on ed h i s for m ula in a l et t er t o Goldba c h in 1750, and th e n p r o v e d it for co nv ex p olyhe dra in 1752. Ku r at o wsk s o b ser v at i o n s ca p t u re d all non planar graphs, and i n 1930 he pub li shed a pr o of of h i s w el l k n o wn graph p l an a r i t y criterion [ W es01]. In 1852, G u t h er i e found t h e famous F our Color problem [ Mi t 81 ]. E v en t h o u gh t h e F our Color p r ob l e m w as i n v e n t ed in 1852, it w as sol v ed only after a c e n t u ry b y Kenn e th Ap p el and W o l fg an g Ha k en [AH77]. F u r t h er mor e, t h e F our Color prob lem has b een c hanged t o F our Color t h eo re m; t h e last w ord on t h e F our Color p r ob l e m has n o t b een said a n ymore. T h i s ti m e i s considered as t h e b i r t h of Graph Theor y [Deo74]. 2.2 Ba s i c Structures and Defini ti ons In o rd e r t o s i mp li fy t h e r ea d i n g, w e h a v e t o st at e t h e b as i c standard n o t at i o n t h at w e will use t h r ou g h o u t t h e w ork. W e will use th e n ot a ti o n g i v e n in te xt bo ok nt r o duction to G r aph Th e b y D o ugl a s B. W est [ W es01] and r ete Mathematics with A b y M. Al b er t so n and J. Hu t c h i n s on [AH88].

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7 2.2.1 Gra phs Definition 1 [ W es01] A g r aph G = (V, E) is a c ombinatorial obj e ct c om p os e d by a p air, whe r e V is the set of verti c es and E is the set of e dges. We c onsider a g r aph to b e simple if E is not a multiset. Usua l ly, the set of verti c es is la b el e d to r ep r esent g r aphs with p oints as verti c es and lines linking these p oints as e dges. See fi g u r e 2.1 [ W es01] for si mp l e graph G wi t h V = { 1, 2, 3, 4, 5, and 6 } and E = { (1,2), (1,5), (2,3), (2,5), (3,4), (4,5), ( 4, 6 ) } Figure 2.1: S i mp l e Graph G n = | V | and e = | E | d eno t e t h e n u m b er of v e rt i ce s and edges res p ecti v el y Note th a t : Th e set of edges t h at co n ta i n s v i s E ( v ). Th e v e r ti c es t h a t sha r e an edge will b e ca ll ed neigh b ors and th e set of n ei g h b o r s of v d eno t ed b y N ( v ). Definition 2 [ W es01] The d e g r e e of a vertex v is the num b er of verti c es adja c ent to v (or e quivalently, the num b er of e dges incident with v). We denote the d e g r e e of v by d e g(v, G) or d e g(v) .The d e g r e e s e quen c e of a g r aph is the s e quen c e form e d by ar r anging the vertex d e g re es in d e c re asing o r der. In fi g u r e 2.1, t h e degree of t h e v er t i ces i s as foll o ws: d eg (1 ) = 2, d e g( 2 ) = 3, d eg ( 3) = 2, d eg (4 ) = 2, d eg ( 5) = 3, and d eg (6 ) = 1. And t h e d eg r ee sequence = { 3 3, 2, 2, 2, 1 } Theorem 1 ( Ha nds ha ki ng Theorem) [ W es0 1 ] If V(G) = { v 1 v 2 . v n } then

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8 P ro of Th e sum of t h e degrees cou n t s t h e to t al n u m b er of t i m es an edge i s inci d e n t wi t h a v er t ex S i n c e t h e degree of a v ertex i s th e n u m b er of edges i ncid e n t wi t h t h at v er t ex Also, e v ery edge i s i ncid e n t wi t h t w o v er ti c es, ea c h edge gets co u n t ed t w i ce, one a t ea c h end. T h u s t h e sum of t h e d eg r ees e qu a l s t wi ce th e n u m b er of edges. Definition 3 [AH88] A g r aph H is a su b g r aph of a g r aph G, denot e d by H G, if V(H) V(G) and E(H) E(G ). Figure 2.2 [AH88] sh o ws H subgraphs of graph G Figure 2.2: Graph G and Subgraph H Definition 4 [AH88] A n i n de p en de n t set of verti c es in a g r aph is a set of mutua l ly non adja c ent verti c es. The inde p enden c e num b er of a g r aph G is the maximum c a r dinality of an inde p endent set of verti c es. It is denot e d by (G ). 2.2.2 Common F a m il ie s of Gra phs Definition 5 A g r aph in which every p air of distinct verti c es is join e d by an e dge is c a l l e d c omplete g r aph A c omplete g r aph with n verti c es is c a l l e d an n c li qu e and is denot e d by K n Figure 2.3 sh o ws ex a mp l es of co mp l et e g raph s. Theorem 2 A n n clique g r aph has exactly n (n 1)/2 e dges. P ro of In K n ea c h v er t ex has d eg r ee n 1. T h u s t h e sum of th e degrees eq u a l s n ( n 1). By (Ha nd sha k i n g Theorem) t h i s sum a l so equals 2 | E | T h u s 2 | E | = n ( n 1) and E = n ( n 1) / 2.

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9 Figure 2.3: Ex a mp l es of Co mp l et e Gr a ph s Definition 6 A g r aph G is c a l l e d c omplete bi p artite g r aph if V(G) has a p artition to two subsets X and Y such that e ach e dge (u,v) G c onn e cts a vertex of X and a vertex of Y. In this c ase, (X,Y) is a bi p artition of G, and G is (X ,Y ) bi p ar tite. Figure 2.4 [AH88] sh o ws c om p l et e b i p a r ti t e graph K 2 3 whi c h has t w o b i par t i t i on subsets X = 2 and Y = 3. Figure 2.4: Co mp l et e Bi p a r t i te Graph K 2 3 Definition 7 A cycle g r aph is a g r aph that c onsists of a single cycle, or in other wo r ds, some num b er of verti c es c onn e ct e d in a clos e d chain, denot e d C n whe r e n is the num b er of verti c es. The num b er of verti c es in C n e quals the num b e r of e dge s, and every vertex has d e g re e 2; that is, every vertex has exactly two e dges incident with it. 2.2.3 Graph O p er a ti ons Th e union of t w o graphs i s formed b y taking th e union of th e v er ti c es and edg es of t h e graphs. Th e join G H of t h e graph G and H i s ob t a i n ed from t h e graph union G H and adding an edge b e tw een ea c h v er t ex of G and ea c h v e rt ex of H

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10 Th e pr o duct G = G 1 G 2 has V ( G 1 ) V ( G 2 ), and t w o v er t i ces ( u 1 u 2 ) and ( v 1 v 2 ) of G are a d ja ce n t if and only if ei t h er u 1 = u 2 and u 2 v 2 E ( G 2 ) or u 2 = v 2 and u 1 v 1 E ( G 1 ). V ertex Rem o v al : If v i i s a v ertex of a graph G = ( V E ), th e n G v i i s t h e i n duced subgraph of G on t h e v er t ex set V v i ; t h at is, G v i i s t h e graph ob t a i n ed after re m o v i n g from G t h e v er t ex v i and all t h e edges i n c i d e n t on v i Edge Rem o v al : If e i i s an edge of a graph G = ( V E ), th e n G e i i s t h e subgraph of G t h at r e su l t s after rem o ving from G t h e edge e i Note t h at t h e end v er t i ces of e i are not rem o v ed from G 2.2.4 Planar Gr a phs Planar graphs a r e t h os e t h a t can b e dr a wn in th e p l a n e so no t w o edges cr oss ex cep t p o ssi b l y a t th e e nd p oi n t of t h e edges. Gr a ph s ar i si n g in ma n y a pp li ca t i on s are planar b y d efi n i t i on su c h as maps of co u n tr i es Ot h er s are planar b y acc i d e n t li k e t r ees. [ S k i 08 ] Definition 8 [AH88] A g r aph G is planar if the r e exists a d r awing of G in the plane in which no two e dges inters e ct in a p oint other than a vertex of G. R eca ll wh a t K 4 l o ok s li k e (Square wi t h edges crossing in t h e ce n t er so t h a t a ll v er t i ces a r e a d ja ce n t ) see fi g u r e 2.3. K 4 has edges whi c h i n t er sec t at n o n v er t ex l o cations. Th er efo r e in i t s original s ta t e K 4 i s n o t planar, bu t K 4 i s i so m o rp h i c to t h e graph b el o w [AH88], whi c h i s planar. Th er ef or e K 4 i s planar. Figure 2.5: A K 4 Planar G raph A s i mp l e, connected, planar graph sp li t s t h e o r i e n t ab l e su r fa ce i n t o a n u m b er o f r eg i on s, including t ot a ll y enclosed r eg i on s and one i n fi n i t e external r eg i on Eu l er

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11 obser v ed a relationship b e tw een t h e n u m b er v of v er t i ce s, th e n u m b er e of e d ge s, and th e n u m b er f of r egi o n s (faces) in su c h a graph [Ger98]. Th i s r el a ti o n sh i p i s kn o wn as E u l er s for m ula. Theorem 3 ( Eul s F or m ula) If a finite, c onn e ct e d, planar g r aph is d r awn on genus g of orientable surfa c es without any e dge inters e ctions, and v is the num b er of verti c es, e is the num b er of e dges and f is the num b er of fa c es, then v e + f = 2 2 g P ro of b y i ndu c ti o n: If G i s ac yc li c, t h en f = 1, and t h e t h eo r em hold s b eca u se t h en G i s a tr ee and e = v 1. O th e rw i se G has a cycle. Let x b e an edge i n a cycle. Deleting x from G and i t s planar dr a wing r es u l ts in a graph G wi t h v v er t i ces, e 1 edges and f 1 faces (since d e l et i n g an edge i n v ol v ed in a cy cl e merges t h e t w o faces on ei t h er side of i t ) By i ndu c ti on w e h a v e v ( e 1) + ( f 1) = 2 2 g and so v e + f = 2 2 g Eu l er s for m ula b y itself d o es n o t p r o v i d e us wi t h a t o ol for sh o w i n g t h at so m e graphs do n o t e m b ed on a su r fa ce of ge n us g b ecause it r efe rs t o t h e set of faces f i n a p r os p ect i v e p l a nar dr a wing of t h e graph. B u t w e can use it to d er i v e th e foll o wing su ffi ci e n t p ri n c i p l e for n on p l an a r i t y Le t ( g ) d eno t e t h e Eu l e r c har a ct er i st i c of ge n us g It i s w e ll k n o wn th a t ( g ) = 2 2 g for or i e n t a b l e su r fa ces. Corollary 1 Sup p ose G i s a connected planar graph, wi t h v n o des, e edges, and f faces, where v 3. Then e 3( v ). P ro of Th e sum of t h e d eg r ees of t h e faces i s equal t o t wi ce t h e n u m b er of edg es from (Ha nd sha k i n g Theorem). B u t ea c h face m ust h a v e degree 3. So w e h a v e 3 f 2 e Eu l er s for m ula s a ys t h at v e + f = 2 2 g so f = e v + and t h u s 3 f = 3 e 3 v + Co m bining t h i s with 3 f 2 e w e get 3 e 3 v + 2 e So e 3( v ). Lemma 1 In any orientable g r aph G with Euler cha r acteristic and thickness t the r e is a vertex of d e g re e at most | (6 6 ) t | P ro of F rom Corollary 1, t h e maxi m um n u m b e r of edges in ea c h planar l ay er i s a t most e 3( v ). If G i s an o r i e n t ab l e graph wi t h Euler c h a r act er i st i c and V v er t i ces has th i c kn e ss t t h en ea c h l ay er has a t m os t 3( v ) edges. In t ot a l G has a t m os t 3( v ) t edges. T h us, e 3( v ) t F rom h a nd sha k i n g t h eorem ( t h eorem 1), w e can fi nd t h e a v erage d eg re e:

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12 Hence, at l ea st one v e r te x has a degree of | (6 6 ) t | or less. In a dd i t i on w e can t el l whether a gi v en graph i s p l an a r or not b y a helpful fact t h a t t h e t w o graphs g i v en in figure 2.6 are b oth non planar. These are t h e Ku r at o wi sk i graphs. T h u s a n y non planar graph m ust co n tain a subgraph closely r el a t ed to one of these t w o graphs. (a) K 5 (b) K 3 3 Figure 2.6: Ku r a t o wsk s G raph s K 5 and K 3 3 Theorem 4 ( Kur a t o ws k s Theorem) a g r aph is planar if and only if it d o es not c ontain any su b division of K 5 or K 3 3 as a su b g r aph. See [ W es01] for th e pr o of of Ku r at o wsk s Th e or em 2.2.5 r Inflated Gr aphs Definition 9 [ABG10] [ABG11] L et G b e a g r aph; the r inflation of G is the lexi co g r aphic p ro duct G [ K r ] and is denot e d by G[r]. G[2] is c a l l e d the clone of G. See fi g u r e 2.7 for an e x am p l e of r i n fl a t ed graph. By d efi n i t i on w e ob t a i n G [ r ] b y re p l aci n g ea c h v ertex of G b y K r and ea c h e d ge of G b y K 2 r (whi c h co n ta i n s a K r for ea c h v ertex of t h e edge). An r i n fl a t i on of G has t h e foll o wing p r o p er ti e s [ABG10] [ABG11]:

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13 (a) K 3 (b) K 5 (c) 3 inflated graph of K 5 Figure 2.7: Ex a mp l e of r i n fl a t i on G r ap h s (a) If th e n u m b er of v er t i ces and edges of G are V and E r es p ec ti v el y t h en th e n u m b er of v er ti c es and edges of G [ r ] are r V a nd r 2 V + r 2 E r es p ec ti v el y (b) G [ s r ] = ( G [ s ]) [ r ]. F or a n y com p l e te graph K s and a n y p o si t i v e i n t eg er r w e h a v e K s [ r ] = K sr (c) Inde p endence i s i n v a ri a n t under i n fl a t i on Th a t is, if t h e i nd e p en d en ce n u m b er of G i s t h en t h e i nd e p en d e n ce n u m b er of G [ r ] i s as w ell. (d) If t h e c li qu e n u m b er of G i s t h en t h e cl i q u e n u m b er of G [ r ] i s r (e) If t h e c hromatic n u m b er of G i s t h en th e c hrom a ti c n u m b er of G [ r ] i s at most r ( f ) A n y edge of G i ndu ces a K 2 r in G [ r ]. 2.3 Thi c k ne ss of a G ra ph By d efi n i t i o n, if a graph i s planar t h en it can b e e m b edded in a si n g l e p l an e Assume t h a t w e are g i v en a n o n p l a n a r graph. H o w ma n y p l an es a r e necessary i n o rd e r t o fully e m b ed i t ? Th i s i d ea l ea d s t o t h e n ex t d efi n i t i o n. Definition 10 The thickness of a g r aph G, denot e d by (G) is the minimum num b er of planar su b g r aphs whose union is G and is a m e asu r e of its d e g re e of non planarity. Determining t h e t h i c k n ess of a graph i s NP co mp l et e p r o b l em [ Mi t 81 ]. Th e graph classes wi t h w ell kn o wn t h i c k n esses a r e t h e c om p l et e graphs, c om p l et e b i par t i t e graphs, and h y p ercu b es. Th e t h i c k n ess of c om p l et e graphs K n i s sol v ed for al most all v alues of n b y Beine k e and Harar y A decade after th a t Alekseev and Gon c ha ko v, so l v ed t h e r em ain i n g cases [ AM S 96 ].

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14 6 Theorem 5 [ S k i 08 ] F or c omplete g r aphs, K n ) = n +7 w ith the ex c eption that K 9 ) = K 1 0 ) = 3 See fi g u r e 2.8 [AMS96] for a d eco m p o si t i on of K 9 i n t o t h r ee planar s ubgr a ph s. Figure 2.8: D eco m p os i ti o n of K 9 2.4 Graph C ol or ing G r aph c oloring i s one of t h e classic p r o b l em s in graph t h eor y It has se v e ra l appli ca t i on s su c h as s c h ed u li n g r eg i st er i n g a ll o cat i o n in co mp il er s, as si gn i n g frequency in Mo b il e r a d i os, etc. A graph m a y b e co l o re d in se v er a l w a ys. W e m a y color ei t h er t h e v e rt i ce s, t h e edges, or t h e faces. W e will f o cus on coloring t h e v e rt i ce s. T h e r u l e i s t o assign a color to ea c h v er t ex in su c h a w a y t h a t t w o connected v e rt i ce s do n o t h a v e th e same color. Therefore, t h e graph i s p r o p erly c olo re d graph and t h e mini m um n u m b er of c ol o r s used t o color a graph i s cal l e d t h e ch r omatic num b er Among prob l em s d ea li n g wi t h co l o ri n g th e most famous one i s ca ll ed F our Color p r oblem whi c h asks if it i s p o ssi b l e t o co l o r a n y planar graph wi t h a t most four col ors. Th i s p r ob l em w as set o u t b y F r a nci s Gu t h r i e in 1852 [ Mi t 81 ], he n o t i ced t h a t four c ol o r s a r e su ffi ci e n t t o co l or t h e map of cou n t i es of England. In 1976, Kenneth Ap p el and W o l fga n g Ha k en pub li sh e d a first pr o of [AH77]. Their pr o of w as b ase d in ruling o u t a l o t of co n fi gu r a t i on s using a c om pu t er b ecause it w as t h e only wa y t o do it in a r easo n ab l e t i m e. D esp i t e of t h i s, it i s assumed t h a t t h e pr o of i s correct.

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15 A graph can b e constructed from a n y map, t h e r eg i on s b e i n g r ep r ese n t ed b y t h e v er t i ces of t h e graph and t w o v er t i ces b eing joined b y an edge if t h e regions corre s p o nd i n g t o t h e v er t i ces a r e a d jac e n t Th e r esu l t i n g graph i s planar. Therefore, a map i s c or r es p o nd s t o a planar graph. Th e F our Color Theorem emphasizes th a t a n y map can h av e i t s faces co l or ed with a t most four c ol o r s su c h t h at no t w o fa ces t h at sha r e a p i ece of b oundary h av e t h e same color. See fi gu r e 2.9 for a 4 colored map of t h e Un i t ed S t a t es. Figure 2.9: F o u r co l or ed m a p Definition 11 A g r aph G is k c olo r able if we c an assign the c olors { 1, 2, 3, . k } to the verti c es in V(G), in such a way that every vertex gets exactly one c olor and no e dge in E(G) has b oth of it s end p oints c olo re d with the same c olor. If k is the sma l lest num b er such that G admits a k c oloring, we say that the ch r om at ic num b er of G is k, and write ( G ) = k Example 1 I l lust r ate how the a b ove definition works: (a) Th e c omplete g r aph K n wi t h n v er t i ces has c hromatic n u m b er n T o see t h at ta k e a n y graph of K n and l o ok a t a n y v e rt ex v i s connected to e v e ry o th e r v e rt ex it can n o t b e t h e same color as a n y other v er t ex Th er ef or e m u s t h a v e a d i ffer e n t color than e v ery o t h er v ertex, whi c h fo r ces n colors. S ee figure 2.10 for an ex am p l e for 8 c h r o m at i c c om p l et e graph K 8 (b) Th e empty g r aphs whi c h a graph wi t h no edges, its c h r o m at i c n u m b er i s 1. S i n ce w e can color all v er t i ces with th e same color. Th e e mp t y graphs are t h e only graphs wi t h c h r om a t i c n u m b er 1. Simpl y a n y graph wi t h an edge n eed s a t l ea st t w o c ol o r s t o pro p erly color i t as b o t h e nd p oi n t s of t h at edge c an n o t b e t h e same co l or. S ee figure 2.11 for an ex am p l e for 1 c h r o m at i c em p t y graph wi t h fi v e v er t i ces.

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16 Figure 2.10: 8 Ch r om a t i c Co mp l et e Graph K 8 Figure 2.11: 1 Ch r om a t i c Em p t y Graph (c) Th e bi p artite g r aphs b y d efi n i t i o n, e v ery b i par t i t e graph wi t h a t l eas t one edge has c h r om a t i c n u m b er 2. S i n ce t h ese graphs a r e co n t ain t w o sets. S ee figure 2.12 for an ex am p l e for 2 c h r o m at i c b i p a r t i te graph K 3 4 Figure 2.12: 2 Chrom a t i c Bi p a r ti t e Graph K 3 4

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17 Theorem 8 L et G b e any g r aph with maximum vertex d e g re e d. Then G is p r op erly (d + 1) c olo r able. P ro of Assume th a t all graphs on k or fe w er v er t i ces wi t h maxi m um degree d are ( d + 1) colorable. Consider G a graph wi t h maxi m um degree d on k + 1 v er t i ces. Find v e rt ex V of degree d in G and delete i t ; call t h e new graph G By induction, w e can pro p erly color G wi t h ( d + 1) or fe w er colors. Th e n r ep l a ce V and its edges t o get G ba c k. Color V wi t h a n y color from among th e ( d + 1) w e started wi t h t h at has n o t b een used on t h e n e i g h b o r s of V T h us, b y induction G i s ( d + 1) colorable. Definition 12 A g r aph is k criti c al if it is k ch r omatic but every p r o p er su b g r aph c an b e p r o p erly c olo re d with fewer than k c olors. 2.5 Vi si bil i t y Repr es e n t at io n for Gra ph Th e i d ea of r ep r ese n t i n g a graph u s i n g a visibility r ep r esentation (VR) w a s i n t r o du ce d in t h e 1980s as a m o d e l t o ol for VL S I wi r e r ou t i n g and ci r cu i t b oard l a y o u t [TT86]. In t h e VR for a graph, t h e v er t i ces map t o o b je ct s in Eu cl i d ea n space and t h e edges are det er m i n ed b y cer t ain v i si b ili t y r el a ti o n s. 2.5.1 Ba r Vi s ibi li t y Gr a phs Bar visibility g r aph for a graph G i s r ep r ese n ted b y h o r i zo n t a l li n e seg m e n t s, su c h t h at t h e v er t i ces of G are re p re se n t ed b y n o n o v e rl a pp i n g hor i zo n t al se gm e n t s ca ll ed v ertex se gm e n t s, and if th e c or r es p o nd i n g se gm e n t s ( a d jac e n t v er t i ces ) a r e v er t i cal l y vi s i b l e, w e can s a y t h a t th e graph i s a bar r ep r es e n t ab l e [ D EG + 05 ]. Fig u r e 2.13 sh o ws an ex am p l e of a b a r v i si b ili t y r ep r ese n t a ti o n wi t h t h e graph it i n duces. Th i s form w as i n t r o du c ed b y Luccio, Mazzone and W ong in 1983 [ LMW87]. Figure 2.13: B ar Vi si b ili t y Graph

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18 2.5.2 Recta ng le Vi s ibi li t y Gr aphs Th e r e ctangle visibility g r aph for a graph G i s r ep r es e n t ed b y dr a wing a graph in t h e p l an e so t h a t t h e v er t i ces of t h e graph a r e th e re ct an g l es and th e ed ge s are hor i zo n t al or v er t i cal li n e seg m e n t s. W e s a y th a t t h e graph i s a r ect a ngl e s v i si b ili t y graph if it has a v i si b ili t y dr a wing. F i gu r e 2.14 [Bei97] sh o ws an ex am p l e of a r ec ta n g l e v i si b ili t y r ep r ese n t a t i on wi t h t h e graph it i ndu ces [ BDHS97]. Figure 2.14: R ect a n g l es Vi s i b ili t y Graph

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19 3. Review of Related Literature In t h i s c hapter, w e r ev i ew some existing t h eor i es and p r ob l em s on graph coloring, and vi s i b ili t y r ep r es e n t at i o n for g r ap h s Th e c h a p t er i s divided i n t o t w o s ect i on s In s ect i o n 3.1, w e b ri e fl y st at e th e cu rr e n t l y famous t h eor et i ca l r esu l t s regarding t h e graph coloring problem. In se ct i on 3.2, w e st at e th e w ell kn o wn t h eo r et i cal r esu l t s c on ce rn i n g t h e v i si b ili t y r ep r ese n ta t i o n p r ob l em for graphs in 2 d i m en si o n s and 3 d i m en si o n s. Th i s i s done to g i v e a frame w ork in whi c h can b e c on si d e re d for those t w o p r ob l e m s. 3.1 Graph Coloring Pr oblem 3.1.1 I n tr o duct io n Let G ) to b e t h e th i c kn e ss of graph G A graph G has thic k ness t wi t h res p ect t o t h e su r fa ce S if G can b e decom p osed i n t o t subgraphs and no fe w er t h an t su c h th a t ea c h of whi c h can b e e m b edded on S T h i s par t i t i on ma k es t co p i es of t h e v e rt i ce s of G and ea c h edge of G i s assign t o one of t h e t co p i es. Mo r e o v er ea c h subgraph of G sho u l d b e e m b ed d a b l e on t h a t su r fac e. Th e c hrom a ti c n u m b er of a graph G d eno t ed ( G ), i s t h e mini m um n u m b er of co l or s needed t o color t h e v er t i ces of G su c h t h a t no t w o ad ja ce n t v er ti c es r ecei v e t h e same color. Let ( g ) d eno t e th e Euler c h a r a ct er i st i c for ge n us g of or i e n t a b l e su r fac es. Th e sphere i s ge n us g = 0, th e to r u s i s ge n us g = 1, and t h e d o ub l e t o r u s i s ge n us g = 2. Let G t,g d eno t e an or i e n t a b l e graph wi t h t h i c k n es s t and ge n us g It i s w ell kn o wn t h at ( g ) = 2 2 g for o r i e n t ab l e su r fa ces [JT95]. See figure 3.1 for ge n us of o ri e n ta b l e su r fa ces. Figure 3.1: Or i e n t a b l e S u r fa ces of ge n us g = 0, 1, and 2 In 1959, Ringel as k ed: Wh a t i s t h e c hromatic n u m b e r of G 2 0 a n y t h i c k n es s t w o and ge n us 0 o r i e n t ab l e graph? [Rin59]. Th e q u est i o n in ge n er al:

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20 What is the b est up p er b ound of the ch r omatic num b er of G t,g any thickness t and any genus g orientable g r aph? 3.1.2 Rela t ed St udie s Th e F our Color th e or em has b een i n v est i g at ed in ma n y r ese ar c h pa p ers and mo t i v at ed n um er ou s coloring prob l em s study whi c h a r e still a li v e and t h e so u r ce of c h a ll eng i n g p r o b l em s and t h eor i es as illustrated b y Jensen and T oft [ JT95]. Al l t h ese p r ob l e m s h av e a basic q u est i o n: how to c olor a g r aph in such a way that two adja c ent verti c es do not sha r e a c olor? F or a ce n t u r y and a half, t h e F our Color p r ob l e m has b een pl ay ed th e foremost r ol e in t h e d e v el o pm e n t of graph t h eo r y It d a t es ba c k t o 1852, when Th o m as Gu t h er i e w as trying t o color th e map of c ou n t i es of England; he n o t i ced t h a t four co l or s suffice. He as k ed h i s b r o th e r F r ed e ri c k Gu t h e ri e if a n y map can b e colored using only four co l o rs so no ad ja ce n t regions colored with t h e same color. Th e n, F rederi c k ex p l ain e d t h e problem t o h i s t ea c h er August DeMorgan, who in t u r n sh o w ed it to Ar t h u r C a yle y Th e problem a t t h e b e ginn i n g pub li sh e d as a pu zzl e b y C a yley in 1878. La t er in 1879, A. B. K em p e g a v e t h e fi rs t published pr o of of t h e F our Color Conjecture. In 1890, He a w o o d p o i n t ed o u t a ser i o u s fl a w i n Kem p e s pr o of. A ce n t u r y later, Ap p e l and Ha k en so l v ed t h e F our Color prob lem [ Mi t 81 ] [AH77]. A t th e t i m e in 1970s, th e pr o of b y Ap p el and Ha k en has n o t b een fully a cce pted. Their pr o of w as based in ruling ou t d i ff er e n t t y p e s of graphs, ( c onfigu r ations) us ing a co mpu t er b ecause it w as t h e only w a y to do it in a se n si b l e t i m e. In s p i te of t h i s, it w as assumed t h at t h e pr o of i s correct [ Mi t 81 ]. R ece n t l y four m a t h em at i ci an s at Ohio S ta t e Un i v er si t y and Ge or g i a I n st i tu t e of T e c hnology (Ro b ertson, Sanders, Seymour and Thomas), g a v e a new pr o of o f t h e F our Color Theorem for a planar graphs. Their pr o of i d ea i s similar t o Ap p el and Ha k e n s pr o of. They found a set of 633 co n fi g ura t i on s and pr o v ed t h a t none of t h em can ap p ear in a sm a ll er c ou n t er ex a mp l e. T h us, t h ey pr ov ed t h at no coun t er ex am p l e ex i st s [ R SS T96]. Actuall y co l or i n g graphs on su r fac es has b een st ud i ed e xt en s i v e l y He aw oo d [Hea90] sh o w ed t h at ea c h graph e m b edded in su c h a su r f ace of ge n us g 1 has c hromatic n u m b er a t most: 48 g +1 2 J

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21 Th i s up p e r b ound i s kn o wn as t h e He a w o o d n u m b er H ( g ). T a b l e 1 sh o ws t h e function of H ( g ) for a ge n us g [ SS 06 ]. T a b l e 1: Th e fu n ct i o n s H ( g ) for a su r fa ce of ge n us g g 1 2 3 4 5 6 7 8 9 10 11 12 H ( g ) 6 7 7 8 9 9 10 10 10 11 11 12 Later, in 1968 Ringel and Y oungs [ R Y68] found t h e c or r es p o nd i n g l o w e r b ounds, b y pr o ving th a t t h e co mp l et e graph on H ( g ) v e rt i ce s can b e e m b edded on a n y su r fa ce of Eulerian ge n us g wi t h t h e ex cep t i on of th e Klein b o tt l e, where th e cor rect b ound on t h e c hrom a ti c n u m b er i s 6 ( He a w oo d s for m ula g i v e s 7) as v e ri fi e d b y F ranklin [ F ra34]. Th e Empi r e p r oblem a l so kn o wn as t h e ( M p i r e p r ob l e m ) asks for t h e maxi m um n u m b er of co l or s needed t o color co u n tr i es su c h t h a t no t w o co u n tr i es sharing a common b order h av e t h e same color where ea c h co u n t r y consists of M d i sjo i n t re gions. He aw oo d [Hea90][JR85] sh o w ed in 1890 t h at th e M p i r e c h r o m at i c n u m b er for a n y ge n us wi t h Euler c har a ct er i st i c wi t h t h e ex cep t i on of ( M = 1 and = 2, F our Color t h eo re m ) at most: He a w o o d [Hea90] sh o w ed t h a t 6 M co l or s are su ffi c i e n t and for t h e case M = 2 t h e b ound w as sharp. F u r t h er mor e, in 1981 H. T a ylor found t h at t h e He a w o o d s b ound w as sharp for M = 3 [Gar05]. Ja c kson and Ringel sh o w ed t h a t 6 M co l or s are su ffi ci e n t for all M > 1 [JR85]. Th e Earth M o on p r oblem w as so coined b y Ringel, whi c h i s a s p ecial case of t h e M p i r e p r ob l e m for co u n t r i es wi t h M = 2 d i sjo i n t r eg i on s wi t h one region of ea c h co u n t ry lying on t h e Ea r t h and one on t h e M o on. In particular, it i s t h e sea r c h for t h e l a rg es t c h r om a t i c n u m b er of a n y graph G 2 0 ( t h i c k n ess t w o and ge n us 0). Ringel [Rin59] r em a r k e d t h a t t h e few c ol o r s w e need t o color all graphs G 2 0 i s li es b e tw een 8 and 12. Th e up p e r b ound 12 comes from a s tr a i g h tf or w a r d i n duction a r gu m e n t t h at i s based on Eu l er s for m ula for p l a n e graphs. Th e l o w er b ound 8 i s due t o th e fact t h a t K 8 has t h i c k n ess t w o as Ringel indicated [JT95]. In 1974, S u l a n k e sh o w ed t h at th e 9 c hrom a ti c j oin of K 6 and C 5 has th i c kn e ss t w o Later, Bo u t i n, Ge th n e r, and S u l a n k e es ta b li sh e d i n fi n i t el y ma n y 9 c hrom a ti c cr i t i ca l graphs of t h i c k n ess t w o [BGS08] [GS09].

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22 In fact, R i n ge s Earth M o on Problem can b e gen er a li zed t o other su r fa ces and more t h an t w o t h i c k n ess It i s w ell kn o wn t h a t G t, 0 i s 6 t c ol o r ab l e. H o w e v er, when t 3 w e do not kn o w if co mp l et e graphs h av e th e maxi m um c hromatic n u m b er among all graphs of t h i c k n ess t Ex cep t if t 3, t h en K 6 t 2 i s t h e largest co mp l et e graph wi t h t h i c k n ess t [BW83]. F or a g i v en ge n us g th e t h i c k n ess t of a graph G i s t h e mini m um n u m b er of t em b ed d a b l e graphs on g for whi c h t h e union i s G If g i s t h e p r o ject i v e p l an e to r u s, o r d o ub l e t o ru s th e n t h e t h i c k n ess of K n i s ( n + 5) / 6 ( n + 4) / 6 and ( n + 3) / 6 r es p ec ti v el y [Bei97]. Le t B ( g ) denote th e maxi m um n u m b er of v er t i ces of a com p l et e graph t h at i s b i e m b ed d a b l e on t h e o r i e n t ab l e su r fa ce g E u l er s for m ula i mp li es t h a t : 73+96 g 2 J Accordingl y T ab l e 2 sh o ws t h e maxi m um o rd e r B ( g ) of a co mp l et e graph t h at i s g b i e m b ed d ab l e for g 3. T a b l e 2: Th e fu n ct i o n s B ( g ) for a su r f ace of ge n us g g 0 1 2 3 B ( g ) 8 13 14 16 In fact, for a n y gi v en graph G 2 1 whi c h can b e e m b edded on t h e t o r u s wi t h G ) = 2, th e mini m um d eg r ee of a v er t ex i s l ess than or equal t o 12, t h u s ( G 2 1 ) 13. Th i s comes from a st r ai g h t fo r w ard induction ar g um e n t t h at b a sed on Eu l er s for m ula for toroidal graphs wi t h ge n us g = 1. Ja c kson and Ringel [ JR00] pr o v ed it and S u l a n k e [ S u l 05 ] sh ow ed t h at K 13 i s b i e m b ed d ab l e S u l an k e found 22 graphs wi t h 13 v er t i ces whi c h can b e e m b edded on t h e t o ru s and whose comple m e n t s can a l so b e e m b edded on t h e t oru s, see figure 3.2 for an ex a mp l e [ S u l 05 ].

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23 Figure 3.2: Bi e m b ed d i n g of K 13 3.1.3 Co nclus io n W e h av e st ud i ed t h e c ol o r i n g p r o b l em in detail. W e h a v e seen t h a t t h e F our Color t h eo re m pr o v ed after a ce n t u r y from it w as defined. W e h av e l o o k e d a t t h e Hea w o o d s n u m b e r and M p i r e problem. More o v er, w e h a v e seen t h a t 6 M co l or s a r e su ffi ci e n t for all M > 1 [Hea90]. W e a l so studied t h e M p i r e s p ec i al case whi c h i s Ea r t h M o o n problem and i t s gen er a li zat i o n for higher t h i c k n ess and d i ff er e n t su r fa ces. In summar y th e E ar t h M o o n problem and its gen e ra li za t i on t o higher t h i c k n ess and d i ffer e n t su r fa ces it has not b een sol v ed y et, ex cep t for t h e t o r oid a l graphs on a surface wi t h Euler c h a r act er i st i c 0 [JR00] [ S u l 05 ]. 3.2 Vi si bil i t y Repr es e n t at io n Pr oblem 3.2.1 I n tr o duct io n Visibility r ep r esentations (VR) of graphs map v er ti c es t o sets in Eu cl i d ea n spa ce and ex p r ess edges as v i si b ili t y r el a t i on s b e tw een t h ese sets. F or graphs G = ( V E ) t h e 3 d i m en s i on a l v i si b ili t y re p re se n t at i o n b y r ect an g l es define as an ar r an g em e n t of d i s joi n t r ect an g l es in R 3 su c h t h at t h e p l a n es det er m i n ed b y t h e rectangles a r e v er t i cal t o t h e z a xi s and t h e sides of th e r ec ta n g l es are parallel t o th e x a xe s and y axes. A g i v e n graph G i s sa i d t o b e r ep r ese n t a b l e if and only if i t s n v er t i ces can b e ass o ciated wi t h n d i sjo i n t r ect an g l es p a r al l e l t o t h e x and y axes i n R 3 su c h t h a t v e rt ex v i and v j are ad ja ce n t in G if and only if th e i r c or r es p o nd i n g rectangles R i and R j are z visible.

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24 Th e c u rr e n t b es t up p er b ound on t h e size of t h e l a r ge st cl i q u e K n t h at can b e r ep r ese n t ed b y rectangles i s 50 [BDHS97] and t h e l o w er b ound i s 22 [ B E L + 93 ]. It still a c h a ll eng e t o narr o w t h e gap b e t w een t h e kn o wn up p e r and l o w er b ound for r ep r ese n t a t i on of K n b y r ect an g l es, t h u s th e qu e st i on is: What is the up p er b ound on the size of the la r gest clique that c an b e r ep r esent e d in 3 dimensions? 3.2.2 Rela t ed St udie s Th e p r ob l em of r ep r ese n ti n g graphs has b een st ud i ed ex t en si v el y in t h e li t er a tu r e as a r esu l t t o t h e l a r ge n u m b er of a pp li ca t i on s su c h as VLSI design, CASE t o ol s ci r cu i t b oard l a y ou t and an i m a t i on p r o b l em on 2 d i m en si o n s. See [ TT86] [DH97] for ex am p l es Bar visibility g r aphs (B V G) in 2 d i m en si o n s (also kn o wn as bar re p re se n t ab l e or v i si b l e graphs) w ere i n tr o duced b y Luccio, Mazzone, and W ong [ LMW87], Wi s math [Wis85], and T amassia and T o lli s [ TT86]. In t h i s form v e rt i ce s map to d i sjo i n t h o r i zo n t a l li n e seg m e n t s in t h e plane, and t w o v e rt i ce s are ad ja ce n t i n t h e graph if and only if t h ei r cor r es p on d i n g seg m e n t s a r e v i si b l e in t h e v er t i ca l direction. W e can see t h at b a r v i si b ili t y graphs a r e planar; more ov er t h ey h a v e b een c h a r act er i ze d as t h o se planar graphs t h at can b e dr a wn in t h e p l an e wi t h a ll cu t v er t i ces on a s i ngl e face. In other w ords, a graph has a bar v i si b ili t y r ep r es en t at i o n if and only if it has a planar e m b e dd i n g su c h t h a t all cu t v er t i ces li e on th e ex t er nal face [Wis85] [TT86]. T h e q u es ti on of whether a graph has a bar v i si b ili t y l a y o u t can b e det er m i n ed in li n ea r t i m e [KW89]. Re ctangle visibility g r aph ( R V G) in 2 d i m en si o n s [BDHS97] i s a v i si b ili t y repre se n t a t i on in t h e p l an e in whi c h t h e v er ti c es of t h e graph map t o closed re ct an g l es and th e edges are e xp r es sed b y hor i zo n t al or v er ti c al v i si b ili t y b e tw een t h e r ec t angles. T w o rectangles are only con s i d er ed t o b e v i si b l e t o one an o t h er if t h er e i s a nonzero wi d t h hor i zo n t al or v e rt i ca l band of si g h t b e tw een t h em Ki r k p at r i c k and Wismath [KW89] h av e sh o wn t h a t e v ery planar graph i s a r ect a ngl e v i si b ili t y graph. Hu t c h i n s on Shermer, and Vince [Bei97] sh o w ed t h at t h e maxi m um n u m b e r of edges e in a re ct an g l e v i si b ili t y graph wi t h n v er t i ces is: H ow e v er, t h i c k n ess t w o graphs m a y h av e at most 6 n 12 edges. Dean and Hu t c h i n son [DH97] est ab li s h ed t h e foll o wing b ound for b i par t i t e graphs. F or n 4, a b i p a rt i t e r ect a ngl e v i si b ili t y graph wi t h n v er t i ces has at m os t 4 n 12 edges.

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25 Also, t h ey pr o v ed th a t K 5 5 i s n o t a r ect an g l e vi s i b ili t y graph and com p l e te graph K 8 i s t h e l ar g est com p l e te graph t h a t a dm i t s a v i si b ili t y r ep r ese n ta t i on in t h i s repre se n t a t i on See fi gu r e 3.3. Figure 3.3: K 8 has a r ec ta n g l e v i si b ili t y graph R ece n t l y i n t er es t in finding v i si b ili t y r ep r ese n t a ti o n in 3 d i m en si o n s has r ece i v e d co n si d er a b l e a tt e n t i on A 3 d i m en si o nal v i si b ili t y dr a wing w as i n t r o du ced as a gen er al i z at i o n of th e 2 d i m en si o n s v i si b ili t y dr a wing. Ad d i t i o nal d e v el o pm e n t of b a r v i si b ili t y graphs t h a t w as s t ud i ed in [ BEL + 93 ] i s t h e class of VR r ep r es e n t ab l e graphs, whi c h i s t h e r ep r e se n t at i o n of v er t i ces b y r ec ta n g l es in 3 d i m e n si on s In t h i s r ep r ese n t a t i on ea c h v er t ex of th e graph maps t o a c l ose d re ct an g l e in R 3 and edges a r e expressed b y v er ti c al v i si b ili t y b e tw een rectangles. Th e r ect an g l es rep r ese n t i n g v er ti c es are d i s joi n t co n t ain e d in p l an es v er t i ca l t o t h e z a x i s, and h a v e sides parallel t o t h e x or y axes. T w o v er t i ces a r e ad j ace n t in t h e graph if and only if th e i r c or r es p o nd i n g re ct an g l es a r e v i si b l e in t h e z direction. A l o t of pa p ers a r e f o cused on th e maxi m um s i ze of a co mp l et e graph wi t h 3 d i m en si o n s v i si b ili t y re p re se n t a ti o n b y r e ct an g l es. In 1994, Bose, E v er et t F e k ete, Lubiw, Meijer, Romanik, Shermer and Wh i t esi d e s con s tr u ct ed a r ect an g l e visi bili t y re p re se n t at i o n of th e com p l e te graph K 20 based on t w o main st r u ct u ri n g bl o c ks. A K 9 can b e bu il t b y placing K 4 and K 5 b l o c k s one u p on th e other a s in fi gu r e 3.4. Th e r i g h t edges of t h e K 5 and t h e b o t to m edges of t h e K 4 p r o v i d e t h e v i si b ili t y needed to a c h i e v e a K 9 A K 18 i s constructed as a join of t w o K 9 co n fi gu r a t i on s. One K 9 i s p l aced as t h e d o wn K 9 and t h e o th e r i s p l aced u p on t h e d o wn K 9 Th e up p er K 9 i s an ex ac t co p y of th e d o wn one, bu t flip p ed o v er and ro t at ed 90 Th e t o p edges of t h e d o wn K 9 and th e left edges of th e up K 9 p r o v i d e t h e v i si b ili t y needed t o a c h i e v e K 1 8 Therefore, b y adding t h e upm os t U and d o wnm os t D rectangles w e h a v e a r ec ta n g l e v i si b ili t y r ep r ese n t a t i on of K 2 0 see fi gu r e 3.5 [ BE F + 94 ]. It w as t h e b es t kn o wn l o w er b ound of t h e l a r ges t c li q u e t h at can b e r ep r ese n ted in 3 d i m en si o n s b y r ect an g l es.

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26 (a) K 4 (b) K 5 (c) K 9 Figure 3.4: Re p re se n t i n g K 9 b y K 4 K 5 b l o c k s Figure 3.5: Re p re se n t at i o n of K 20 R ece n t l y Rote and Ze ll e [ FM99] h a v e found a r ect an g l e vi s i b ili t y r ep r ese n ta t i on of K 22 b y using si m u l at ed a nn ea li n g a l g or i t hm. In [CDHM96] it w as sh o wn t h a t no com p l e te graph with more t h an 102 v er t i ces has su c h a re p re se n t at i o n. Th i s r esu l t w as t h en impr o v ed t o 55 in [ BE L + 93 ] b y F e k ete, Houle, and Wh i t esi d e s, t h ey pr ov ed th a t K 56 d o es not h a v e a r ect a ngl e v i si b ili t y r ep r es e n t at i o n. Th ei r pr o of i s based on t h e analysis of unimaximal subsequences in sequences of rect angle c o o rd i n a t es. A sequence x 1 x 2 ... of distinct i n t eg er s i s called unimaximal if it has exactly one l o cal maxi m um, i.e., for all i, j k wi t h i < j < k w e h a v e x j > min { x i x k } E v e n t ual l y b y S to l a [ BD HS 97 ][ S 09 ] it w as reduced t o 50, her pr o of i s a l so based on t h e study of unimaximal subsequences in th e sequences of rectangle c o o r d i n at es bu t she consider ea c h c o ord i n a te d e p en d e n t l y Th er e a r e t w o wa ys in whi c h t w o b o d i es can b e con s i d er ed equal [FHW95]: (a) I sotheti c : t h ey can b e made i d e n t i ca l b y t r a n sl at i o n s onl y (b) Congr uent : t h ey can b e made i d e n t i ca l b y t r a n sl at i o n s and ro t at i o n s.

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27 So, t w o i so th e ti c o b ject s h a v e t h e same size, wh il e t w o c on g r u e n t o b ject s h a v e t h e same sha p e. According t o [FHW95], if t h e v e rt i ce s a r e re p re se n t ed b y un i t squares t h en th e largest com p l e te graph with t h i s t y p e of re p re se n t at i o n i s K 7 See fi gu r e 3.6, t h e six phases i nd i ca t e h o w t h e se v en squares a r e p l a ces on t o p of ea c h o th e r. Figure 3.6: R ep r ese n t i n g K 7 b y un i t sq uar es In summar y F e k ete and Mei j er [FM99] p r o v i d ed th e r esu l t s for re ct an g l e v i si b ili t y r ep r ese n t a t i on s for t h e g i v e n n u m b er of sha p es or sizes. Th e foll o wing T a b l e 3 sh o ws t h e b est kn o wn up p er b ound for t h e gi v en n u m b er of sha p es or sizes; t h e (min) i nd i ca t es t h e b es t kn o wn l o w er b ound and (max) t h e b est kn o wn up p er b ound. R ega r d i n g c om p l e te b i par t i t e graphs, in [ BE F + 94 ] t h e a u t hor s found t h a t th e re i s a r e ct an g l e v i si b ili t y r ep r ese n t a t i on s of co mp l et e b i p a rt i t e graphs K m,n for a n y m and n where m and n are t h e size of ea c h p a rt i t i o n. In fi gu r e 3.7 it i s easy t o see t h at a n y K m,n admits a r ec ta n g l e v i si b ili t y r ep r ese n t a ti o n s. Figure 3.7: A n y K m,n has a r ec ta n g l e v i si b ili t y r ep r ese n t a t i on

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28 T a b l e 3: L ow er and up p er b o und s for r ec ta n g l e v i si b ili t y r ep r ese n t a t i on s Num. of sha p es m i n max Num. o f si ze s m i n max 1 2 3 4 5 6 12 18 20 20 20 22 14 28 42 55 55 55 1 2 3 4 5 6 7 8 9 10 11 7 12 18 20 20 20 20 20 20 20 22 7 14 21 28 35 42 49 55 55 55 50 F u r t h er mor e, t h ey a l so p r o v e d t h at t h er e i s a r ec ta n g l e v i si b ili t y r ep r ese n ta t i on for c om p l et e b i p a r ti t e graphs t h a t a r e for m e d b y rem o ving a p erfect m a t c h i n g d eno t ed b y K n,n M where n i s t h e size of b oth of t h e p a rt i t i o n s. F or pr o of see [ BEF + 94 ]. According t o F e k ete, Houle, and Wh i t esi d es [ FHW95], a n y com p l et e graph K n can b e r ep r ese n t ed b y un i t d i sc s. See fi g u r e 3.8. Figure 3.8: A n y K n has a d i s cs v i si b ili t y r ep r ese n ta t i on

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29 3.2.3 Co nclus io n W e h av e studied th e vi s i b ili t y r ep r e se n t at i o n s of graphs in 2 d i m en si o n s and 3 dimensions. In 3 d i m en si o n s, w e h a v e seen th a t all planar graphs are rectangle v i si b ili t y graph. C om p l et e graphs K n h a v e r ect a ngl e v i si b ili t y r ep r ese n t a t i on for n 22. R eg ard i n g co mp l et e b i p a r t i te graphs, w e h a v e kn o wn t h at K m,n h a s a r ect a ngl e v i si b ili t y r ep r ese n t a t i on for all m and n and K n,n m i n u s a p erfect mat c hing a dm i t s r ect a ngl e v i si b ili t y r ep r es e n t at i o n. Finall y w e h av e l o o k ed a t v ar i a n t vi s i b ili t y r ep r es e n t at i o n s b y un i t s qu a r es and d i scs as w ell.

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30 4. Optimization Sear c h Me th o ds Th e m ea n i n g of o p ti m i z at i o n i s finding a parameter in a function t h a t ma k es a b et t er so l u t i on [DB07]. All of appropriate v alues are p oss i b l e so l u t i on s and t h e b est v a l u e i s an o p t i m um so l u t i on T o so l v e o p t i m i zat i o n p r o b l em s, o p t i m i zat i o n a l go r i th m s a r e used. Cat eg o ri z at i o n of o p ti m i zat i o n al g or i t hm can b e p e rf orm ed b y l o oking a t t h e n a t u re of t h e a l go r i th m s and t h i s d i vi d e s th e al g or i t hm s i n t o t w o cat eg o ri e s: deterministic a l go r i t hm, and st o c h a st i c a l go r i th m s [ Bla89]. Deterministic al g or i t hm s foll o w a p r eci se pr o cedure, and i t s p a t h and v a l u es of b oth design v a r i ab l e s and t h e f un ct i on s a r e r e p ea t ab l e S t o c h a st i c a l go r i th m s i n general are ca ll ed heuristic al go ri t hm s. Th e basic i d ea for all heuristic a l g or i t hm s i s t o c h o ose an i n i ti a l so l u t i on and t h en t r y i n g t o i mp r o v e th i s so l u ti o n b y c h o os ing another sol u t i o n t h a t b elongs t o t h e n e i g h b o r h o o d of t h e cu rr e n t sol u t i o n. If some re qu i r em e n t s a r e sat i sfi e d, t h i s new sol u t i o n i s accep t ed t o b e t h e new cur r e n t s olu t i o n. All h e u ri s ti c m e th o d s h a v e a s to pp i n g c on d i t i o n, whi c h m i g h t b e t h e n u m b er of i t er at i o n s, or sh o wing th a t t h er e i s no o p p o r tu n i t y to i mp r o v e t h e cu rr e n t so l u ti o n a n ym o r e [ S k i 08 ] [ P or05]. In t h i s c h a p t er w e will d i s cu ss and l o ok a t four d i ffe re n t heuristic sear c h meth o ds: si m u l a t ed a nn ea li n g a l g or i t hm and t h r ee of th e S w arm I n t el li gen ce al g or i t hm s, whi c h a r e a n t colo n y op t i m i za t i on a l g or i t hm, cu c k o o op t i m i za t i on a l g or i t hm, and firefly al go ri t hm. 4.1 S w arm I n te ll ig ence Algorithms ( SI ) S w arm i n t el li gen ce ( S I) i s based on c ol l ec ti v e b eh a vior of sel f or g an i ze d sy st em s natural ex a mp l es of S I al go r i th m s : P ar t i cl e S w arm Op t i m i za t i on (PSO), A n t Colo n y Op t i m i za t i on ( A CS), Firefly Al g or i t hm ( F A), Cu c k o o Op t i m i za t i on Al g or i t hm (C O A), Bacteria F oraging (BF), th e Ar t i fi ci a l Bee Colo n y (ABC), and so on. It can b e used in co n t ro lli n g ro b ots, p r ed i c ti n g s o cial b eh a v i or s, enhancing th e t el eco m m un i ca t i on and co mpu t er ne tw orks, e t c. Actuall y t h e S I can b e ap p li e d t o a v arie t y of fi el d s in e ngin ee ri n g and com pu t er sc i en ce [PCN10]. In general, a s w arm i n t el li ge n ce algorithm for op t i m i za t i on p r ob l e m s w o rk s wi t h a s w arm of i nd i v i du a l s where ea c h individual cr ea t es one sol u t i o n. Then th e so l u t i on s are impr o v ed heuristically b y using t h e i n for m a t i on a b o u t g o o d so l u t i on s t h at h a v e a l r ea d y b een o b t ain e d b y t h e s w arm. Mo r e o v e r, resear c hers h a v e de v elo p ed m u l t i s w arm v e rs i on s for par t i cu l a r i m p o r ta n ce, for ex am p l e when mor e t h an one o p t i mal so l u t i o n sh ou l d b e returned. [JMM0 8 ].

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31 W e r ev i ew some p opular al go ri t hm s in t h e field of s w arm i n te lli gen c e for prob l em s of op t i m i za t i on as foll o ws: 4.1.1 A n t Colo n y Optimization Algorithm ( A CO ) A n t colo n y op t i m i za t i o n i s a heuristic a l go r i t hm [ DBS06], it i s one of th e most r ece n t te c hn i q u es for a pp r o x i m a t e o p t i m i za ti o n. It i s inspired b y a d ap t a ti o n of a n a t u r al system from r ea l a n t c ol o n i es. In p a r ti c u l ar, A CO i s inspired from t h e fo r ag i n g b eh a v i or of a n ts. When sea r c h i n g for f oo d, a n t s i n i t i a ll y d i sc o v er th e a r ea su rr ou nd i n g th e i r nest in a random manner. As s o on as an a n t fi nd s a f oo d so urc e, it e v a l u at es t h e q u a n ti t y and t h e qu a li t y of t h e f o o d and ca r ri e s some of it ba c k t o t h e n es t. During t h e return trip, t h e a n t d e p os i ts a c h em i c al pheromone tr a il on t h e ground. Th e qu a n t i t y of pheromone d e p osi t ed will g u i d e o t h er a n t s t o t h e f oo d so u r ce. Th i s co m m un i ca t i on b e tw een a n t s via pheromone t r ai l s e n ab l es t h em t o find sh or t est p a th s A CO has b een i n i t i a t ed b y Marco Dorigo [ Dor92] whi c h has b een s u ccess fu ll y applied to se v eral NP hard co m b i n a t or i a l p r o b l em s su c h as v eh i cl e r ou t i n g problem, tr av eling sa l esm a n problem, pr o duction s c h ed u li n g se q u e n t i al o r d er i n g p r o b l em t el ec om m un i ca t i on r o u t i n g, etc. [ GN10]. Figure 4.1 sh o ws t h e a l go r i t hm in simplified m o del as foll o wing steps [Blu05]: (a) All a n t s a r e in t h e nest. Th e re i s no pheromone in t h e e n v i r on m e n t. (b) Th e foraging st ar t s. In p r o b ab ili t y 50% of t h e a n ts ta k e t h e short path (sym b o li zed b y c i rc l es) and 50% t a k e t h e l o n g path t o t h e f oo d so u r ce (sy m b olized b y sq uar es ). (c) Th e a n t s th a t h a v e t a k en t h e sh or t p a t h h a v e arri v ed ea r li er at t h e f o o d so u r ce. Therefore, when returning, t h e p r o b ab ili t y t o t a k e again t h e sh or t path i s higher. (d) Th e pheromone trail on t h e sh or t p a t h r ece i v e s, in prob a b ili t y a stronger r ei n fo r cem e n t and th e prob a b ili t y t o t a k e t h i s p a t h gr o ws. Finall y due t o t h e e v a p or a ti o n of t h e pheromone on th e long path, t h e wh o l e co l o n y will, i n p r ob a b ili t y use t h e sho r t path. Th e d i ffer en c es b e tw een t h e m o del and t h e b eh a v i or of r ea l a n t s i s t h e m o del con sists of a graph G = ( V E ), where V consists of t w o n o des, namely v s r ep r ese n t i n g t h e nest of t h e a n t s, and v d r ep r ese n t i n g t h e f oo d so u r ce. F u r t h er mor e, E co n si st s of t w o links, namely e 1 and e 2 b e t w een v s and v d T o e 1 w e assign a length of l 1 and t o e 2 a l eng t h of l 2 su c h t h at l 2 > l 1 In o t h er w ords, e 1 r ep r ese n t s t h e sho r t p a t h b e t w een v s and v d and e 2 r ep r ese n t s t h e long path. R ea l a n t s d e p o si t pheromone on t h e p at h s on whi c h t h ey m o v e.

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32 P i = Figure 4.1: A n t s find sho rt es t path b e tw een th e i r nest and f oo d sou r ce s T h us, th e c h em i ca l pheromone t r ai l s are m o deled as an ar t i fi ci a l pheromone v a l u e i for ea c h of t h e t w o li n k s e i i = 1 2. Su c h a v a l u e i nd i cat es t h e s tr eng t h of t h e pheromone trail on t h e co rr es p on d i n g path. Ea c h a n t S t ar t s from v s and m o v e wi t h p r o b ab ili t y : i 1 + 2 where i = 1 2 O b viousl y if 1 > 2 t h e p r o b ab ili t y of c h o osing e 1 i s h i gh e r, and v i ce v ersa. F or returning from v d t o v s an a n t uses t h e same p a t h as it c hose t o r ea c h v d and it c hanges t h e ar t i fi ci a l pheromone v a l u e ass o ciated t o t h e used edge. Finall y a ll a n ts con du c t t h ei r return trip and reinforce t h ei r c hosen p a t h. Pseudo C o de Th e b as i c a n t colo n y op t i m i za t i on al go ri t hm ( A CO) can b e su mmar i ze d as t h e pseudo c o de sh o wn in figure 4.2 [DBS06]. Example and Algorithm T ra ce (a) Ex a mpl e: W e will clarify t h e steps of th e A CO algorithm b y an ex a mp l e to find largest i nd e p en d e n t set S of v er t i ces of V in a graph G = ( V E ) su c h t h at for ea c h edge ( x, y ) E ei t h er x / S or y / S In figure 4.3 [Ski08] t h e graph G i s t h e i npu t and b y applying A CO al go ri t hm w e will ge t t h e supset S t h e o u tp u t Th e c h a ll eng e li es n o t in finding an i nd e p en d e n t set, bu t in finding a largest i nd e p e nd e n t set. F u r t h er m o re finding a maxi m um i nd e p e nd e n t set i s NP com p l e te [ S ki 08 ].

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33 Figure 4.2: Pseudo c o de of A CO al go ri t hm Figure 4.3: Ex a mp l e for Applying A CO Al go r i th m Th e steps will b e re p eated u n til t er m i n a t i on c on d i t i o n sa ti s fi ed whi c h i s in t h i s case t h e n u m b er of i t er a ti o n. First p l a ce all a n t s on t h e s ta r t v er t ex (start) l et su b se t S 0 = { sta r t } and i n i ti a li ze pheromone t r ai l s. Then, do t h e foll o wing t o t h e n a n ts: 1 S = S 0 2 F or j = 1 t o th e n u m b er of v e rt i ce s 3 Ch o ose th e n ex t v er t ex (next) t o v i si t acco r d i n g to t h e p r ob a b ili t y P = j If ( sta r t, nex t ) / E t h en S = S + { nex t } 4 U p d a t e th e a m ou n t of pheromone for ea c h n o de acco r d i n g t o t h e l o cal solu t i on 5 If t h e new s olu t i o n S from th i s i t er at i o n i s b et t er than cu rr e n t so l u t i on S 0 t h en re p l ace t h e cu rr e n t so l u t i on wi t h th e new one. If t h e n u m b er of i tera t i on s i s r ea c h to th e maxi m um, t h en te rm i n a t es t h e al go ri t hm; ot h er wi s e go t o step 2.

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34 After t er m i n a ti o n w e will get th e op t i m a l so l u t i on for t h e maxi m um i nd e p en d e n t set S for a g i v en graph G (b) Algorithm T r a ce: Th e a l go r i t hm tr a ce i s for t h e example. Le t t h e n u m b er of i tera t i on s = 2 and t h e n u m b er of a n t s = 3. All a n ts will s ta r t from t h e v ertex start see figure 4.4 (a). Let S 0 = { sta r t } and th e pheromone on ea c h v ertex = 0. F or fi rs t a n t do t h e foll o wing: I n i ti a li ze S t o b e S 0, th e n c h o ose t h e n ex t v er t ex t o v i si t b y t h e prob a b ili t y P whi c h i s e qu a l s t o 0 for ea c h m ov e in t h e fi r st i t er at i o n. T h e first a n t has t w o c hoices t o m o v e wi t h t h e same p r o b a b ili t y a s sh o wn in fi gu r e 4.4 (b), so let c h o ose one of th e m as a next see fi g u r e 4.4 (c ), bu t (sta r e next) i s an edge in t h e graph, so nothing will c hange and c h o ose t h e n ex t v er t ex to vi s i t. F i g u r e 4.4 (d) sh o ws th a t t h e a n t has a l so t w o c hoices to m o v e with t h e same p r o b ab ili t y so l e t c h o ose one of th e m as a next as sh o wn i n figure 4.4 (e), (sta r e, next) i s n o t an edge in t h e graph. T h us, S = S + { nex t } see figure 4.4 ( f ), and so on u n til t h e first a n t finish its t o u r b y r ea c h i n g t h e maxi m um n u m b er of v e rt i ce s. A t t h i s p oi n t, t h e first a n t gi v e us t h e first so l u ti o n. Nex t u p date t h e pheromone trail ( i ncr em e n t b y one) on ea c h v e rt ex whi c h has b een added to t h e S R e p ea t th e same steps t o t h e t w o remaining a n ts for t h e fi rs t i t er at i o n. After t h a t c h o ose t h e b est s olu t i o n and r ep l a ce t h e cu rr e n t so l u t i on wi t h t h e b es t new one and re p eat for t h e second i t er a t i on After t er m i n at i o n w e will get t h e op t i m a l so l u ti o n for t h e maxi m um i nd e p en d e n t set S for a g i v e n graph G

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35 Figure 4.4: T r ac e for Applying A CO Al go r i t hm 4.1.2 Cu ck o o Optimization Algorithm ( C O A ) Cu ck o o op t i m i za t i on al go ri t hm w as de v elo p ed b y Y ang and Dep [ YD09]. It i s a heuristic sea r c h a l go r i t hm i n sp i r ed b y t h e cu c k o o bird b r eed i n g b eh a v i or [ K C11]. Th e cu c k o o bird l a ys her eggs in th e nest of another host s p ecies. Th e host t a k es ca r e of t h e eggs b el i ev i n g t h at t h e eggs a r e i t s o wn. If t h e h os t d i s c o v er s t h a t an egg i s not its o wn, it m a y e i th e r destr o y t h e egg or t h e nest and t h e n build a new nest a t a d i ffer e n t l o cation. Th e cu c k o o b re ed i n g a n a l og y i s used for de v eloping new design o p t i m i zat i o n a l go r i t hm. A gen er at i o n i s re p re se n t ed b y a set of host nests. Ea c h n es t car r i es an egg ( sol u t i o n ) T h e q ual i t y of so l u t i on s i s i mp r o v ed b y gen er at i n g a new s olu t i o n from an existing so l u t i on b y m o difying cer t ain c harac t er i st i cs. Th e new s olu t i o n i s formed b y a random m o v e on t h e selected sol u t i o n. If t h e new s olu t i o n i s found t o b e su p er i o r t o another randomly c hosen existing so l u t i on t h en t h e old so l u ti o n i s r ep l aced wi t h t h e new one. T h us, t h e b est so l u t i on s in ea c h gen e ra t i o n are carried o v er t o th e n ex t gen er a t i on T o start t h e o p t i m i za ti o n a l go r i t hm [ R a j11], ea c h cu c k o o st ar t s l a ying eggs ran domly in some other host b i r d s nests. After all cu c k o os eggs are laid in host b i r d s nests, some of t h em are detected b y host b i r d s and t h ey thr o wn ou t of t h e nest, t h ese eggs h a v e no c hance t o gr o w. Rest of t h e eggs gr o w in host nests, h a t c h and are fed b y host birds. Another i n t er es ti n g p o i n t a b o u t laid cu c k o o eggs i s t h a t only one egg in a nest has th e c hance t o gr o w. Th i s i s b ecause when cu c k o o egg h at c h e s

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36 and t h e c h i c k s come out, she t h r o ws t h e host b i r d s o wn eggs ou t of t h e n es t. In case t h a t host b i r d s eggs hat c h earlier and cu c k o o egg hat c hes later, cu c k o o s c hi c k ea t s m os t of t h e f oo d host bird b r i n gs t o th e nest ( b ecause of her 3 t i m es bigger b o d y she pushes other c h i c k s and eats more). After co up l e of d a ys th e host b i r d s o wn c h i c k s d i e from h unger and only cu c k o o c hi c k re main s in t h e nest. When y oung cu c k o os gr o w, t h ey li v e in t h ei r o wn area and s o cie t y for som e t i me Bu t when t h e t i m e for egg l a ying ap p r oa c h es t h ey i mm i g r at e t o new and b et t er e n v i r on m e n t wi t h more si m il ar i t y of eggs t o host b i r d s and also wi t h more f oo d for new y oungsters. Th i s e n v i ronm e n t i s se l ect ed as t h e goal for o t h er cu c k o o s t o immigrate. After t h at e v alu a t ed th e new e n vi r o nm e n t wi t h t h e p r ev i ou s one and c hose t h e b est as a sol u t i o n. Wh e n all cu c k o os immigrated t o w ard goal p oi n t t h en new egg l a ying p r o c ess restarts. Pseudo C o de Th e basic cu c k o o op t i m i za t i on algorithm (C O A) can b e s ummar i zed as t h e p seu d o c o de sh o wn in figure 4.5 [R a j11]. Figure 4.5: Pseudo c o de of CS a l go r i t hm Example and Algorithm T ra ce (a) Ex a mpl e: W e will explain t h e steps of th e C O A algorithm b y an ex a mp l e to color t h e v er t i ces V in a graph G = ( V E ) using t h e mini m um n u m b e r of c ol o r s s u c h t h at x and y h a v e d i ffer e n t c ol o r s for all ( x, y ) E In fi g u r e 4.6 th e graph G i s t h e i npu t and b y applying C O A al g or i t hm w e will ge t th e ou t pu t colored graph [ S k i 08 ].

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37 Figure 4.6: Ex a mp l e for Applying C O A Al go r i th m T o apply t h e C O A a l g or i t hm, w e r e p ea t t h e foll o wing steps u n t il n u m b er of v er t i ces t h a t pro p erly colored r ea c h t o t h e maxi m um. 1 First i n i t i al i z e so l u ti o n S wi t h a random so l u t i on Then, do t h e n ex t st ep s t o n cu c k o os. 2 G i v e co l or s t o a cu c k o o 3 Let t h e cu c k o o to color v er t i ces 4 uncolored t h e v er ti c es t h a t a r e r eco gn i z ed not pro p erly colored. 5 C ou n t t h e v er t i ces t h at pro p erly colored, and r ed u ce t h e n u m b er of cu c k o os who impro p erly co l o re d v e rt i ce s. 6 E v aluate new s olu t i o n and u p d at e c u rr e n t s olu t i o n S if t h e new so l u t i on i s b et t er t h an curre n t. If t h e n u m b e r of i tera t i on s i s r ea c h t o th e maxi m um, t h en t er m i n at es t h e al go ri t hm; o t h er wi se go t o step 1. After t h e a l go r i t hm t er m i n at es w e will h a v e an o u tp u t as a graph G p r o p e rl y co l or ed with mini m um n u m b e r of c ol o r s (b) Algorithm T r a ce: Th e a l go r i t hm t r ac e i s for t h e e xa mp l e. T h e steps will b e re p eated u n t il t h e n u m b er of v er t i ces t h a t a r e pro p erly co l or ed r ea c h 10. First i n i ti a li ze so l u t i on S wi t h a random so l u t i on as sh o wn in fi g u r e 4.7 (a). Let t h e n u m b e r of cu c k o os = 2. F or t h e four cu c k o os do t h e foll o wing: Gi v e fi rs t color to th e cu c k o os and l e t t h e first one t o color t h e v er t i ces, see fi g u r e 4.7 (b). Then uncolored t h e v er t i ces t h at are re cog n i zed n o t pro p erly co l or ed and co u n t t h e n u m b er of v er t i ces t h a t a r e p r o p e rl y colored, whi c h are (3 v er ti ces) t h e first so l u t i on see fi g u r e 4.7 (c). Then

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38 let t h e second c o c k o o t o color v er t i ces with th e same color randomly as s h o wn in fi g u r e 4.7 (d). Th e n uncolored th e v er t i ces t h at a r e re cog n i zed not pro p erly co l or ed and cou n t t h e n u m b er of v er t i ces t h a t are pro p erly co l o r ed whi c h are (3 v er t i ces) t h e second so l u t i on see figure 4.7 (e). After t h at r ed u ce t h e n u m b er of cu c k o os who impro p erly co l or ed l a r ges t n u m b e r of t h e v er ti c es (whi c h i s t h e second cu c k o o). T h u s th e n u m b er of cu c k o o i s one. E v alu a t e new so l u t i on s and u p d a t e cu rr e n t so l u t i o n as sh o wn in fi g u r e 4.7 ( f ). Re p eat t h e same st ep s wi t h t h e second, t h i r d, fou r t h, ..., n co l or s, u n t il t h e n u m b er of v er t i ces t h at are p r o p e rl y co l o re d rea c h 10. Figure 4.7: T r ac e for Applying C O A Al go r i t hm 4.1.3 Firefly Algorithm ( F A) Th e firefly a l g or i t hm i s a heuristic a l g or i t hm [ Y an09], it inspired b y t h e flashing b eh a vior of fi r efl i e s. Th e f und am e n t a l pur p ose of fi r efl y s flash i s t o act as a signal system t o a tt r a ct o th e r fi re fl i es. T h us, ea c h fi re s m o v em e n t i s based on a b sor p t i o n of t h e others. Th e fl ash i n g li g h t can b e for m ulated in su c h a wa y whi c h ma k es it p o ssi b l e t o i n v e n t new op t i m i za t i o n al go r i th m s [ CM12]. F or si mp li ci t y Y ang for m ulated t h ese fl ash i n g c har a ct er i st i cs as t h e foll o wing r u l es: 1. All fi r efl i es are un i se x, so t h a t one firefly will b e a t tr a ct ed t o all o t h er fi r efl i e s r eg ard l ess of t h ei r s ex

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39 2. A tt ra ct i v en ess i s re l at i v e t o fi r efl y s b r i g h t n ess, so for a n y t w o fi r efl i e s, t h e l ess b r i g h t firefly will b e a tt ra ct ed t o t h e b ri g h t er one and m ov e t ow ard i t H ow e v er, t h e b ri g h t n ess can decrease as t h ei r d i s ta n ce i ncr ea ses If t h er e i s no b ri g h t er one than a gi v en firefl y it will m ov e randoml y 3. Th e b r i g h t n ess of a firefly i s det er m i n ed b y t h e o b j ect i v e function. Pseudo C o de Based on p r ev i o u s t h re e ru l e s, th e basic firefly al g or i t hm ( F A) can b e su mmar i ze d as t h e pseudo c o de sh o wn in figure 4.8 [ Y an09]. Figure 4.8: Pseudo c o de of F A algorithm Example and Algorithm T ra ce (a) Ex a mpl e: W e will d em o n st r at e t h e steps of t h e F A al go ri t hm b y an ex am p l e t o Color th e edges E in a graph G = ( V E ) u s i n g th e mini m um n u m b e r of co l or s su c h t h a t no t w o same color edges share a common v e rt ex In fi gu r e 4.9 t h e graph G i s th e i npu t and b y applying F A al go ri t hm w e will get t h e o u t pu t co l o re d graph [ S k i 08 ].

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40 Figure 4.9: Ex am p l e for Applying F A Al go r i t hm s Th i s a l g or i t hm st a rt s w i t h gen e ra t i n g i n i ti a l p opulation of n fi r efl i es ( v er t i ces ). Determining t h e li g h t i n ten si t y I t o ea c h firefly b y o b j ect i v e function, whi c h i s t h e v er t ex degree in t h i s ex a mp l e. R e p ea t t h e a l g or i t hm u n til t r ea c hes t h e n u m b er of edges e. In i t i a li ze color c wi t h first color and t h en do t h e n e xt st ep s : 1 F or i = 1 t o n ( n u m b er of fireflies) do 2 F or j = i + 1 t o n do 3 T est if ( i, j ) i s an edge in a g i v en graph t h en 4 If th e I j I i and firefly i have not an edge co l or ed wi t h c th e n color t h e edg e b e tw een i and j 5 E v aluate new so l u t i on and u p date li g h t i n t en si t y b y decreasing I i b y on e After finishing t h e a b ov e steps for one i t er a ti o n i ncr em e n t c b y one and r e p ea t u n t il th e a l g or i t hm t er m i n at es. By t h e end w e will h av e a graph G p r o p e rl y colored. (b) Algorithm T r a ce: Th e algorithm t ra ce i s for t h e e xa mp l e. First w e are gen er a ti n g i n i ti a l p opula t i on of n firefly whi c h i s 8 v e rt i ce s. Then w e a r e det er m i n i n g t h e li g h t i n t en s i t y I t o ea c h firefly whi c h i s t h e v ertex degree. All t h e v er t i ces h a v e d eg r ee eq u a l s t o 3 ex cep t t h e v er t ex in t h e m i dd l e has degree eq u a l s t o 7, see fi gu r e 4.10 ( a ). Th e steps will b e r e p ea t ed u n t il t h e n u m b er of edges r ea c h es to t h e maxi m um whi c h i s 14. Next, i n i t i al i ze color c wi t h first color and do t h e n ex t steps t o 8 fi r efl i es: F or firefly i = 1 t es t if t h er e i s an edge b e t w een it and t h e firefly j t h en compare t h e d eg r ee, see fi gu r e 4.10 (b). If t h e degree of v er t ex j i s gr ea t er t h an or eq u a l s t h e d eg r ee of v er t ex i and v ertex i has n o t an edge colored with fi rs t color, t h en color t h e edge b e t w een i and j as sh o ws in fi g u r e 4.10 (c).

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41 Then d ec re m e n t th e li g h t i n ten si t y to t h e firefly i b y one, see fi gu r e 4.10 (d). Do t h e same steps wi t h all j fi r efl i es. After finishing wi t h first firefly do th e same steps for firefly i = 2 as s h o wi n g in fi g u r e 4.10 ( e and f ). R e p ea t t h ese steps t o all i fi r efl i es, t h en after t h e first i t er a ti o n set color c wi t h t h e second color and test t h e stop c on d i t i o n i f it sa t i sfi ed th e n t h e a l g or i t hm t er m i n a te s; ot h er wi s e another i tera t i on will t a k es place. Figure 4.10: T race for Applying F A Al go r i th m s 4.2 Si m ulated Annealing Algorithm ( SA) S i m u l a ted an n ea li n g i s a h eu r i s ti c a l g or i t hm, and it w as d e v e l o p ed b y Ki r k p a tr i c k Ge l at t and V ec c h i [ K GV88]. S i m u l at ed a nn ea li n g inspired from t h e p h y si ca l pro cess of c o o li n g molten m a te ri a l s d o wn t o t h e so li d s ta t e. Theoreticall y t h e e n er gy state of a system i s d e scr i b ed b y t h e energy s ta t e of ea c h p a r t i cl e co n st i t u t i n g i t Th e energy st at e ju mp s randoml y wi t h su c h t ran si t i o n s managed b y th e t em p er a tu r e of t h e system. Th e t r an s i ti o n p r ob a b ili t y P ( e i e j T ) from energy e i t o e j a t t em p er a tu r e T i s g i v en b y: e i e j P ( e i e j T ) = e k B T Wh er e k B i s a co n st a n t [ S k i 08 ].

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42 Th er e are t w o m a jor p r o cesse s t h a t t h e si m u l a t ed a nn ea li n g al g or i t hm m u s t go t h r ou g h. First, for ea c h t em p er a tu r e t h e al go ri t hm r un s t hrou g h a n u m b er of i t er at i o n. Th e n u m b er of i t er at i o n i s determined b y th e programmer. In ea c h i t er at e t h e inputs a r e r a nd om i z ed and once t h e n u m b er of i t er at i o n has b een com pleted, t h e t em p er at u r e can b e l o w e re d. Second, if th e tem p er at u r e i s l o w ered, it i s decided whether or n o t th e t em p er at u r e has r ea c h ed t h e l o w e st t em p er at u r e a ll o w e d. If t h e t em p er a tu r e i s not l o w er t h a n t h e l o w est t em p e ra t u r e, t h en th e t em p er a tu r e i s l o w er ed and an o t h er i tera t i o n of r an d o m i za t i on s will t a k e place. If t h e t em p er a tu r e i s l o w er t h an t h e l o w est tem p erature, t h e si m u l a ted a nn ea li n g algorithm t er m i n a t es. Pseudo C o de Th e basic s i m u l at ed a nn ea li n g a l go r i t hm (SA) can b e su mm a r i zed as t h e p seu d o c o de sh o wn in figure 4.11. Figure 4.11: Pseudo c o de of SA al g or i t hm Example and Algorithm T ra ce (a) Ex a mpl e: W e will clarify t h e steps of t h e SA a l go r i t hm b y an ex am p l e t o find largest cl i q u e S in a graph G = ( V E ) su c h t h at for all x, y S ( x, y ) E In figure 4.12 [Ski08] t h e graph G i s t h e i npu t and b y applying SA algorithm w e will get th e largest cl i q u e S t h e o u tp u t T h e c h a ll eng e li e s n o t in finding a clique, bu t in finding a l ar g est clique. F u r t h er m o re finding a maxi m um cl i q u e i s NP co mp l et e [ S k i 08 ].

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43 2 2 2 Figure 4.12: E xa mp l e for Applying SA Al go ri t hm s First, create i n i t i al s olu t i o n S 0 whi c h i s a subgraph of G wi t h e = n ( n 1) edg es. In i t i a li ze t em p er a tu r e t Th e steps will b e r e p ea t ed u n t il t h e t em p er at u r e r ea c h es zero or th e re i s no b et t er so l u t i on (no subgraph larger t h an S wi t h e = n ( n 1) edges). Then, do t h e fo ll o w s for i = n ( n u m b er of v e rt i ce s) iterations: 1 Generate random sol u t i o n S in t h e neig h b orh oo d of s olu t i o n S 0 and com p a r es t h ei r n u m b er of n o des and e d ges 2 If S i s b e tt er t h a n S 0 t h en S i s accep t ed as a new sol u t i o n; o th e rw i se, S i s accep t ed wi t h a p r ob a b ili t y P T h e h i gh e r t h e t em p e ra t u r e, t h e more li k el y it i s t o a cce p t w orse sol u t i o n s. 3 Decrease t h e t em p er a t u re Nex t decided wh e th e r or n o t th e t em p er at u r e has r ea c h ed th e l o w e st t em p e ra t u r e al l o w ed (zero). If t h e t em p er a tu r e i s n o t zero an o t h er i t er at i o n will t a k e p l a ce; o t h er wi se, t h e a l g or i t hm t er m i n at es After t er m i n at i o n w e m i g h t get t h e so l u t i on for t h e maxi m um c li qu e S for a g i v e n graph G if it found. (b) Algorithm T r a ce: Th e al go ri t hm t r a ce i s for t h e example. First w e generate i n i t i a l random sol u t i on S 0 whi c h i s a subgraph of G wi t h t h e n v er t i ces and e = n ( n 1) S ee fi gu r e 4.13 (a) for t h e so l u ti o n S 0 whi c h i s a subgraph of G wi t h t h e 3 v er t i ces and 3 edges ( K 3 ). Th e n w e ini t i al i z e t h e tem p er at u r e t o 100. Th e steps will b e r e p ea t ed u n t il t h e t em p er at u r e r ea c h es zero or t h er e i s no c hange on th e sol u t i o n. F or th e n u m b er of v er t i ces 17 do th e foll o wing: Generate a random so l u ti o n S from t h e neig h b orh oo d of th e so l u ti o n S 0 and compares th e i r n u m b er of v er ti c es and edges, see fi g u r e 4.13 (b). If so l u ti o n S i s b et t er than so l u t i on S 0 t h en S i s accep t ed as a new s olu t i o n; o t h er wi se S i s accep t ed wi t h t h e p r ob a b ili t y P whi c h i s de p ends on t h e

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44 d i ffer en c e b e t w een S and S 0 and t h e t em p er at u r e. Th e higher t h e t em p e ra t u r e, t h e more li k e l y t o accep t w o r se sol u t i o n. T h u s so l u ti on S i s as same as s olu t i o n S 0 and w e accep t ed as a new sol u ti o n as sh o wn in fi gu r e 4.13 (c). After 17 i t er at i o n s decrease t h e t em p er at u r e b y 0.96 and another i tera t i on will t a k e p l a ce u n t il t h e t em p er at u r e r ea c h es ze r o or t h er e i s no c hange it t h e so l u t i on S Figure 4.13: T race for Applying SA Al go r i th m s

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45 5. Resear c h Met h o do lo gy Th i s resear c h w as conducted in order t o co m b i n e h e u ri s ti c sea r c h meth o d s wi t h graph t h eo r et i c m e th o d s to so l v e t w o c l ass i cal p r ob l em s in graph th e or y (Graph co l or i n g p r ob l em and cer t a i n vi s i b ili t y r ep r ese n ta t i on s prob l em for graphs). F i r st t h ey are used t o generate graphs wi t h s p ec i fi c c hromatic n u m b ers and finding i t s t h i c k n ess, and v i ce v ersa. T h e goal of creating and exploring t h ese graphs i s t o d i sc o v er more a b o u t t h e b o und s of t h e c hromatic n u m b er for th e Earth M o on p r ob l e m and i t s ge n er al i za t i o n t o higher t h i c k n ess and d i ffer e n t su r fa ces. S e con d, t h ey a r e used t o r ep r ese n t t h e p r o b l em of v i si b ili t y in graph form and t o d i sc o v er more a b ou t t h e b ou nd s of t h e l a rg est size of com p l e te graphs t h at h a v e a r ec ta n g l es v i si b ili t y re p re se n t at i o n in 3 d i m en si o n s. In order t o a c h i e v e t h ese r es ear c h goals, w e c hose th e th r ee foll o wing t e c hn i q u es: 5.1 Fir s t T e c hniq ue W e used a t e c hn i q u e i n sp i r ed b y t h e n e xt pro p o si t i on T o t h at end, let G r ep r ese n t t h e co mp l em e n t of G Pro p osition 1 [GS09] Sup p ose G is a g r aph with n verti c es. If G then is K m f r e e, P ro of Gi v en a graph G th e v e rt i ce s b eing colored wi t h t h e same color form an i nd e p en d e n t set, whi c h in t u r n form a co mp l et e subgraphs in t h e c om p l em e n t Hence, if G dose not co n t a i n K m t h en at most m 1 v e rt i ce s of G can b e colored wi t h t h e same color, whi c h l ea d s: In t h i s t e c hn i q u e, w e s ta r t with a graph G of kn o wn t h i c k n ess t on ge n us g and find i t s c h r o m at i c n u m b er b y e li m i n at i n g t h e n u m b er of K m subgraphs of t h e com p l em e n t of t h e union of t subgraphs u n t il it r ea c h es zero. T h i s t e c hn i q u e i s ap p li e d b y using t h r ee d i ffer e n t o p ti m i z at i o n a l g or i t hm s (A n t Co l o n y Op t i m i zat i o n, F i r efl y Al g or i t hm, and S i m u l a t ed An n ea li n g ).

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46 m \ \ l \ l \ l \ l I m plem e n t a ti on In order to ma k e th e co mp l em e n t G of G K free and pr o duce n c ri t i ca l m 1 graph wi t h n v er t i ces, w e p erformed t h e foll o wing steps for b oth th e A n t Colo n y Op ti m i z at i o n and Firefly Al g or i t hm s: 1. Generate t t r i an g u l at i o n s on ge n us g b y using S u l a n k s T r so f t w a re 2. Find an i n i t i al so l u t i on b y randomly flipping t h e d i ago nal s of qu a d r il at er a l s in t h e t r i an g u l at i o n. 3. Co u n t t h e n u m b er of K m subgraphs on t h e com p l e m e n t of t h e union of t t r i an g u l at i o n s. 4. Im p r o v e t h e cu rr e n t so l u t i o n b y randomly flipping t h e d i ago n a l s of quadri l a tera l s of t h e t r i an g u l a ti o n s u n t il t h e co mp l em e n t of t h ei r union has no K m 5. T es t ea c h remaining edge of t h e t ri a n g u l at i o n s one at a t i m e, r em o v e a n y edge whi c h l e a v es t h e c om p l e m e n t of t h e union wi t h no K m 6. F orm t h e union G of t h e t t r i an g u l at i o n s. 7. If a n y v ertex of G has degree l ess t h a n n 1 t h en r ejec t G ; o th e rw i se, m 1 8. Che c k t h a t t h e graph G i s n c hrom a ti c b y using Dh a r w a d k e s s of t w a r e m 1 V ertex Coloring A lgorithm [Dha11]. 9. T es t ea c h r em a i n i n g edge of G one a t a t i m e, r em o v e a n y edge whi c h l e a v es G n c hrom a ti c m 1 10. If G has l ess t h a n n v er t i ces t h en reject G ; o th e rw i se, 11. G i s n o w n c ri t i ca l wi t h n v er t i ces. m 1 H o w e v er, in th e si m ulated an n e al i n g al go ri t hm w e p erformed exactly t h e sam e steps with a small d i ffer en ce In step four w e find a new random so l u t i on in t h e neig h b orh oo d of t h e cu rr e n t so l u ti o n and c om p a r es t h ei r v a l u es If th e new so l u t i on i s b e tt er t h an t h e cu rr e n t t h en th e new sol u ti o n i s accep t ed as th e n e w cu rr e n t so l u ti o n; ot h er wi s e t h e new so l u t i on i s accep t ed wi t h t h e p r ob a b ili t y th a t de p ends on t h e d i ff er en ce in t h e o b ject i v e fu n ct i on s and on t h e t em p e ra t u r e. Th e h i g h er t h e t em p e r at u r e, t h e more li k ely it i s t o accep t w o r se l o ca l s olu t i o n s. Wh il e in t h e p r ev i ou s al g or i t hm s w e just accep t ed t h e b et t er l o ca l so l u t i on s.

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47 ( G ) ( G ) l 5.2 Second T e c hni que W e used a t e c hn i q u e i n s p i re d b y th e n ex t p r o p os i ti o n s. Pro p osition 2 L et G b e any g r aph. Then ( G ) |V ( G ) | P ro of Gi v en a k coloring of G th e v er t i ces b eing co l or ed wi t h th e same color form an i nd e p e nd e n t se t. Let G b e a graph with n v er t i ces and c a k coloring of G W e define V i = { v | c ( v ) = i } for i = 0 1 ..., k Ea c h V i i s an i nd e p en d e n t set. Let ( G ) b e t h e i nd e p en d e n ce n u m b er of G w e h a v e V i ( G ). S i n ce n = | V ( G ) | = | V 1 | + | V 2 | + ... + | V k | k ( G ) = ( G ) ( G ) w e h av e: ( G ) \ |V ( G ) | Pro p osition 3 Sup p ose G is a g r aph with Euler cha r acteristic on n verti c es and e e dges, then the thickness of G P ro of It i s w ell kn o wn t h at a planar graph on ge n us g wi t h n 3 can h a v e no more than 3( n ) edges. A maximal planar graph i s one t o whi c h no edges can b e added w i th o u t d est r o y i n g p l a nar i t y Gi v en a n o n p l a nar graph G a maximal planar subgraph H of G i s a subgraph of G t o whi c h no edges of G H can b e added wi t h ou t destr o ying p l an a r i t y A maxi m um planar subgraph i s one whi c h has maxi m um n u m b er of edges among all planar subgraphs of G Th e t h i c k n ess G ), of a graph G i s th e mini m um n u m b er of planar subgraphs whose union i s G whi c h l e ad s: W e s ta r t wi t h a graph G on ge n us g of kn o wn h i gh e r c h r om a t i c n u m b er according t o P r o p o si t i on 2 and find i t s t h i c k n ess b y finding subgraphs whose union i s G ea c h subgraph has enough edges t o h av e t h i c k n ess t ill u st r at ed in Pr o p o si t i on 3. T h i s t e c hn i qu e applied using cu c k o o o p ti m i z at i o n a l g or i t hm, a general r an d o m i ze d heuristic approa c h for finding g o o d s olu t i o n s for op t i m i za t i on p r ob l e m.

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48 ( G ) I m plem e n t a ti on In t h i s i mp l em e n t at i o n, w e found t h e d ec om p o si t i on of a gi v en graph G i n t o t subgraphs whi c h can b e e m b edded on an or i e n t a b l e ge n us g b y t h e foll o wing steps: 1. Generate a graph G 2. Che c k th a t t h e graph G i s \ |V ( G ) | c hrom a ti c b y using D har w a d k er s so f t w a r e V ertex Coloring A lgorithm [Dha11]. 3. Random planar graphs ( t ri a n g u l at i o n s) of t h e selected ge n us are gen e ra t ed wi t h at most 3( n ) edges and a t most 2( n ) faces, and do n o t co n t a i n a n y s u b d i vi s i on of K 5 or K 3 3 as illustrated in Ku r a t o ws k s Theorem 4. 4. Randomly i n t er c h a ngin g d i ag on a l to minimize th e n u m b er of edges n o t i n t h e t r i an g u l a ti o n s bu t are in th e graph b eing decom p osed u n t il t h er e i s no edge t h at dose n o t b elongs t o a n y t r i an g u l a ti o n s. 5. N o w, G i s decom p osed i n t o t subgraphs. 5.3 Third T e c hniq ue W e consider sequences of n rectangles lying parallel t o th e x, y p l a n e in R 3 Th e sequence i s valid if its ass o ciated v i si b ili t y graph i s K n Ea c h r ec t an g l e R in a v alid sequence can b e d escr i b e in t er m s of t h e p er p en d i cu l a r d i st an c es from com mon p o i n t O t o ea c h of i t s sides. I n st ead of giving t h e x, y c o o r d i n at es of R w e d esc ri b e R as a 4 t up l e ( E r N r W r S r ) whose c o o r d i n a te s gi v en r es p ect i v el y t h e d i st a n ces from O R t o t h e east, n o r t h, w est and sou t h sides of R W e assume th a t ea c h c o o rd i n a t e v a l u e of ea c h of t h e n rectangles i s a p o si t i v e i n t eg er in r a n ge [0 n ] w i th o u t c h a n g i n g t h e v i si b ili t y r el at i o n sh i p s among th e rect angles. Sup p ose t h a t t w o re ct an g l es in a v alid sequence, A = ( E a N a W a S a ) and B = ( E b N b W b S b ), t h e i n t er sect i o n of t h e i r p r o je ct i on s o n to t h e x, y p l a n e d eno t ed b y A B Then A B co n t ain s O and th e c o o r d i n at es of A B are E = min { E a E b } N = min { N a N b } W = m i n { W a W b } and S = m i n { S a S b } W e s a y t h a t a corner of A B i s f r e e if it i s n o t c ov ered b y a n y of t h e pr o jec ti o n s of r ec ta n g l es o ccurring b e tw een A and B in th e sequence [ BEF + 94 ]. In summar y th e rectangle A and B can see ea c h o t h er if and only if one of t h e foll o wing f r e e c orner co nd i t i on s F C h o l d s for all t h e re ct an g l es R b e tw een A and B [ BE L + 93 ]:

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49 F C n e ( A, B ) n o rt h ea st i s free, i.e. ( E r < min { E a E b } or N r < min { N a N b } ) F C nw ( A, B ) nor t hw es t i s free, i.e. ( N r < min { N a N b } or W r < min { W a W b } ) F C sw ( A, B ) so u t hw est i s free, i.e. ( W r < min { W a W b } or S r < min { S a S b } ) F C s e ( A, B ) so u t h eas t i s free, i.e. ( S r < min { S a S b } or E r < min { E a E b } ) Th e re p re se n t at i o n of K n i s found u s i n g th r ee d i ffer e n t o p t i m i zat i o n a l go r i t hm s (A n t Colo n y Op t i m i za t i on Cu c k o o Op t i m i za ti o n Al go ri t hm, and Firefly Algo r i t hm ). I m plem e n t a ti on F or g i v en n t h e a l go r i th m s try t o find a r ea li za ti o n of K n as foll o ws: 1. Generate an i n i t i al random so l u t i on 2. Co u n t t h e n u m b er of rectangles needed t o b e rem ov ed from th e collection in order t h a t ea c h r ect an g l e can see t h e ot h er s ( o b je ct i v e fu n c ti o n ) Th a t i s det er m i n ed in a straig h tfor w ard manner from th e free corner c on d i t i o n s F C 3. Im p r o v e t h e cu r re n t so l u ti on b y randomly se l ect i n g t w o rectangles and s w ap ping one of t h e four c o ord i n a t es u n t il t h e o b j ect i v e function rea c hes zero. 4. N o w, K n i s r ep r ese n ta b l e wi t h rectangles in 3 dimensions.

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50 6. Fi nding s and D iscussi ons In t h i s c h ap t er w e conclude th i s study b y sh o wing our r esu l t s and d i sc u ssi n g so m e r eco mm en d a ti o n s for fu t u r e w ork. 6.1 Res ult s of the Study In t h i s stud y w e h a v e f o cused on t w o p r ob l em s r el e v a n t t o t h e co m binatorial p r ob l e m s in graph t h eo ry : (i) Graph co l or i n g problem (ii) Vi si b ili t y r ep r ese n t a t i on for graph in 3 d i m en si o n s Th e aim of t h i s t h esi s w as t o i mp r o v e t h e w ell kn o wn up p er b ound for t h e c hro m a ti c n u m b er of a n y ge n us g t h i c k n ess t o ri e n ta b l e graph, and t h e up p er b ound on th e s i ze of th e l ar g est cl i q u e t h at can b e r ep r ese n t ed in 3 d i m en s i on s b y m ea n s of t h e heuristic sea r c h meth o d s In order t o do t h at w e t r i ed t o ans w er se v e ra l q u est i o n s ( see r ese ar c h q u es ti o n s) Th e r es u l ts w ere as foll o w: First: D o es the r e exist a g r ap h G 2 0 with ( G 2 0 ) = 10, 11, or 12? T o ans w er t h e first q u es ti o n, w e h av e t o sh o w t h a t t h e up p er b ound of th e c hro m a ti c n u m b er for a n y ge n us 0 and t h i c k n es s t w o graph i s 12, t h en w e apply t h e first t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p o si t i o n 1. Th e Euler c h a r act er i st i c for o r i e n t ab l e su r fa ces g i s ( g ) = 2 2 g so (0) = 2. Th e M p i r e c h r om a t i c n u m b er for a n y su r fa ce wi t h Euler c haracteristic i s at most [Hea90][JR85]: ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (2 pire, 2) (6 2)+1+ (6 2+1) 2 24(2) 2 12. Corollary 2 If 23 | V ( G ) | 24 and G is K 3 f r e e then ( G ) 12 If 34 | V ( G ) | 36 and G is K 4 f r e e then ( G ) 12 If 45 | V ( G ) | 48 and G is K 5 f r e e then ( G ) 12

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51 F or ( K m n ) { ( K 3 23) ( K 4 34) ( K 5 45) } w e h a v e n o t y et succeeded on finding 1 2 c h r om a t i c t h i c k n ess t w o graph on t h e sphere. Th e mini m um n u m b e r of com p l et e graphs K m in t h e co mp l em e n t s w ere 93, 69, and 162, r es p ec t i v e l y b y applied A CO, F A, and SA a l g or i t hm s. Corollary 3 If 21 | V ( G ) | 22 and G is K 3 f r e e then ( G ) 11 If 31 | V ( G ) | 33 and G is K 4 f r e e then ( G ) 11 If 41 | V ( G ) | 44 and G is K 5 f r e e, then ( G ) 11 F or ( K m n ) { ( K 3 21) ( K 4 31) ( K 5 41) } w e h a v e not y et succeeded on find ing 1 1 c h r om a t i c t h i c k n ess t w o graph on t h e sphere. T h e mini m um n u m b er of co mp l et e graphs K m in th e co mp l em e n t s w ere 35, 106, and 140, r es p e ct i v e l y b y applied A CO, F A, and SA a l g or i t hm s. Corollary 4 If 19 | V ( G ) | 20 and G is K 3 f r e e then ( G ) 10 If 28 | V ( G ) | 30 and G is K 4 f r e e then ( G ) 10 If 37 | V ( G ) | 40 and G is K 5 f r e e then ( G ) 10 F or ( K m n ) { ( K 3 19) ( K 4 28) ( K 5 37) } w e h a v e n o t y et succeeded on finding 1 0 c h r om a t i c t h i c k n ess t w o graph on t h e sphere. Th e mini m um n u m b e r of com p l et e graphs K m in t h e com p l e m e n t s w ere 5, 71, and 43, re s p ect i v el y b y applied A CO, F A, and SA a l g or i t hm s. Corollary 5 If 17 | V ( G ) | 18 and G is K 3 f r e e then ( G ) 9 If 25 | V ( G ) | 27 and G is K 4 f r e e then ( G ) 9 If 33 | V ( G ) | 36 and G is K 5 f r e e, then ( G ) 9

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52 F or ( K m n ) { ( K 3 17) ( K 4 25) ( K 5 33) } w e h a v e succeeded on finding 9 c hromatic th i c kn e ss t w o graph on th e sphere. See Ap p endix B, it c o n t a i n s 90 new 9 cr i t i ca l graphs. T h e graphs w ere selected from a c at a l og u e of h und re d s found, where t h e ca nd i d a te s w ere c hosen for th e un i q u en e ss of degree s eq u en ces. Figure 6.1 sh o ws an ex a mp l e of new 9 cr i ti c al t h i c k n es s t w o graph on 17 v er t i ces whose com p l e m e n t i s K 3 free, figure 6.2 sh o ws an ex am p l e of new 9 cr i t i ca l t h i c k n ess t w o graph on 25 v er t i ces whose c om p l em e n t i s K 4 free, and figure 6.3 i s a new 9 cr i t i cal th i c kn e ss t w o graph on 33 v e rt i ce s whose c om p l em e n t i s K 5 free. Figure 6.1: New 9 C ri t i ca l Graph whose Co mp l em e n t i s K 3 f re e

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53 Figure 6.2: New 9 C ri t i ca l Graph whose co mp l em e n t i s K 4 f re e Figure 6.3: New 9 C ri t i ca l Graph whose co mp l em e n t i s K 5 f re e Second: D o es the r e exist a g r ap h G 3 0 with ( G 3 0 ) = 17 or 18? T o ans w er t h e second q u est i o n, w e h av e t o sh o w t h a t t h e up p er b ound of c h r om a t i c n u m b er for a n y ge n us 0 and t h i c k n ess t h r ee graph i s 18, t h en w e apply t h e fi r st

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54 ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (3 pire, 2) (6 3)+1+ (6 3+1) 2 24(2) 2 18. Corollary 6 If 35 | V ( G ) | 36 and G is K 3 f r e e then ( G ) 18 If 52 | V ( G ) | 54 and G is K 4 f r e e then ( G ) 18 If 69 | V ( G ) | 72 and G is K 5 f r e e then ( G ) 18 F or ( K m n ) { ( K 3 35) ( K 4 52) ( K 5 69) } w e h a v e n o t y et succeeded on finding 1 8 c h r om a t i c t h i c k n ess t h r ee graph on t h e sphere. Th e mini m um n u m b e r of com p l et e graphs K m so far in t h e c om p l e m e n t s w ere 371, 606, and 754, r es p e ct i v e l y b y ap p li e d A CO, F A, and SA a l go r i t hm s. Corollary 7 If 33 | V ( G ) | 34 and G is K 3 f r e e then ( G ) 17 If 49 | V ( G ) | 51 and G is K 4 f r e e then ( G ) 17 If 65 | V ( G ) | 68 and G is K 5 f r e e, then ( G ) 17 F or ( K m n ) { ( K 3 33) ( K 4 49) ( K 5 65) } w e h a v e not y et succeeded on find ing 1 7 c h r om a t i c t h i c k n ess t h r ee graph on t h e sphere. Th e mini m um n u m b er of co mp l et e graphs K m in t h e co mp l em e n t s w ere 307, 332, and 696, r es p e ct i v el y b y applied A CO, F A, and SA a l g or i t hm s. Third: D o es the r e exist a g r a ph G 4 0 with ( G 4 0 ) = 23 or 24? T o ans w er t h e t h i rd q u es ti o n, w e h av e t o sh o w t h a t th e up p er b ound of c hromatic n u m b er for a n y ge n us 0 and t h i c k n es s four graph i s 24, t h en w e apply t h e fi r st t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p osi t i o n 1 Th e Euler c h a ra ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (0) = 2. Th e M p i r e c hrom a ti c n u m b er for a n y su r fa ce wi t h Euler c h a r ac t er i st i c i s a t most:

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55 ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (4 pire, 2) (6 4)+1+ (6 4+1) 2 24(2) 2 24. Corollary 8 If 47 | V ( G ) | 48 and G is K 3 f r e e then ( G ) 24 If 70 | V ( G ) | 72 and G is K 4 f r e e then ( G ) 24 If 93 | V ( G ) | 96 and G is K 5 f r e e then ( G ) 24 F or ( K m n ) { ( K 3 47) ( K 4 70) ( K 5 93) } w e h a v e not y et succeeded on find ing 24 c h r o m at i c t h i c k n ess four graph on t h e sphere. Th e mini m um n u m b er of co mp l et e graphs K m in t h e co mp l em e n t s w ere 236, 351, and 880, r es p e ct i v el y b y applied A CO, F A, and SA a l g or i t hm s. Corollary 9 If 45 | V ( G ) | 46 and G is K 3 f r e e then ( G ) 23 If 67 | V ( G ) | 69 and G is K 4 f r e e then ( G ) 23 If 89 | V ( G ) | 92 and G is K 5 f r e e then ( G ) 23 F or ( K m n ) { ( K 3 45) ( K 4 67) ( K 5 89) } w e h a v e not y et succeeded on find ing 23 c h r o m at i c t h i c k n ess four graph on t h e sphere. Th e mini m um n u m b er of co mp l et e graphs K m in t h e co mp l em e n t s w ere 198, 359, and 459, r es p e ct i v el y b y applied A CO, F A, and SA a l g or i t hm s. F ourth: D o es the r e exist a g r ap h G 3 1 with ( G 3 1 ) = 19 ? T o ans w er t h e fo u r t h q u est i o n, w e h a v e to sh o w t h at th e up p er b ound of c h r om a t i c n u m b er for a n y ge n us 1 and t h i c k n es s th r ee graph i s 19. Then w e apply t h e se con d t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p osi t i o n 2 and Pr o p osi t i o n 3. Th e Euler c h a ra ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (1) = 0. Th e M p i r e c hrom a ti c n u m b er for a n y su r fa ce wi t h Euler c h a r ac t er i st i c i s a t most:

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56 ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (3 pire, 0) (6 3)+1+ (6 3+1) 2 24(0) 2 19. Th e graph K 19 whi c h has 171 edges i s th e union of t h re e graphs ea c h of whi c h can b e e m b edded on ge n us 1. S i n c e t h e graph t h at has n v er t i ces has a t most 3( n ) edges can b e e m b edded on a su r fac e wi t h Euler c har a ct er i st i c t h en t h e graph K 19 can b e e m b edded on t h e to r u s ( ge n u s 1) whi c h has 19 v er ti c es has a t most 57 edges. T h u s K 19 i s a graph wi t h ( G 3 1 ) = 19. By applying second t e c hn i qu e and t h e S u l a n k e s T so f t w a re w e found t h e d eco m p os i ti o n of G 3 1 = K 19 i n t o t h r ee subgraphs ( G 3 1 = G 1 U G 2 U G 3 ) whi c h can b e e m b edded on ge n us 1 as foll o ws: Figure 6.4: D eco m p os i ti o n of K 19 Fifth: D o es the r e exist a g r ap h G 4 1 with ( G 4 1 ) = 25 ? T o a n s w e r t h e fifth q u est i on w e h a v e t o sh o w t h a t t h e up p er b ound of c hrom a ti c n u m b er for a n y ge n us 1 and t h i c k n ess four graph i s 25. Th e n w e apply t h e seco nd t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p osi t i o n 2 and Pr o p osi t i o n 3. Th e Euler c h a ra ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (1) = 0. Th e M p i r e c hrom a ti c n u m b er for a n y su r fa ce wi t h Euler c h a r ac t er i st i c i s a t most: 6 M +1+ (6 M +1) 2 24 ( M pire, ) 2 (6 4)+1+ (6 4+1) 2 24(0) Hence, (4 pire, 0) 2 25.

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57 Th e graph K 25 whi c h has 300 edges i s t h e union of four subgraphs ea c h of whi c h can b e e m b edded on ge n us 1. S i n ce th e graph t h a t has n v er t i ces has at most 3( n ) edges can b e e m b edded on a su r fac e wi t h Euler c h a r a ct er i st i c t h en t h e graph K 25 can b e e m b edded on t h e to r u s ( ge n u s 1) whi c h has 25 v er t i ces has a t most 75 edges. T h u s K 25 i s a graph wi t h ( G 4 1 ) = 25. By applying second t e c hn i qu e and t h e S u l a n k e s T so f t w a re w e h a v e not succeeded on finding t h a t K 25 has t h i c k n ess four. Th er e w as just one edge t h a t d o es n o t b elongs t o a n y subgraphs. Th e graph K 25 has a t most t h i c k n ess fi v e as sh o wn in fi gu r e 6.5. Figure 6.5: D eco m p os i ti o n of K 25 Bu t b y applying t h e same meth o d s w e found th e d e com p o si t i o n of G 4 1 = K 2 4 whose ( K 2 4 ) = 24 i n t o four subgraphs ( G 4 1 = G 1 U G 2 U G 3 U G 4 ) whi c h can b e e m b edded on ge n us 1 as foll o ws:

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58 Figure 6.6: D eco m p os i ti o n of K 24 Sixth: D o es the r e exist a g r ap h G 2 2 with ( G 2 2 ) = 14 ? T o ans w er t h e si x t h qu e st i on w e h a v e t o sh o w t h a t t h e up p e r b ound of c hromatic n u m b er for a n y ge n us 2 and t h i c k n ess t w o graph i s 14. Then w e apply t h e seco nd t e c hn i q u e, whi c h i s i n sp i r ed b y p r o p o si t i o n 2 and p r o p o si t i on 3. Th e Eu l e r c h a r a ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (2) = 2. Th e M p i r e c h r om a t i c n u m b er for a n y su r fa ce wi t h Euler c haracteristic i s at most: ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (2 pire, 2) (6 2)+1+ (6 2+1) 2 24( 2) 2 14. Th e graph K 14 whi c h has 91 edges i s th e union of t w o graphs ea c h of whi c h can b e e m b edded on ge n us 2. S i n c e t h e graph t h at has n v er t i ces has a t most 3( n ) edges can b e e m b edded on a su r fac e wi t h Euler c har a ct er i st i c t h en t h e graph K 14 can b e e m b edded on t h e d o ub l e t o r u s ( ge n u s 2) whi c h has 14 v er ti c es has a t most 48 edges. T h u s K 14 i s a graph wi t h ( G 2 2 ) = 14. By applying second t e c hn i qu e and t h e S u l a n k e s T so f t w a re w e found th e d ec om p o si t i o n of G 2 2 = K 14 i n t o t w o subgraphs ( G 2 2 = G 1 U G 2 ) whi c h can b e e m b edded on ge n us 2 as foll o ws:

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59 1. G 1 T r i a n gu l a t i on : 0: 2 13 10 1: 2 6 3 11 8 5 10 13 7 4 2: 0 10 9 7 8 11 6 1 4 13 3: 1 6 5 7 13 9 11 4: 1 7 9 13 2 5: 1 8 7 3 6 12 9 10 6: 1 2 11 9 12 5 3 7: 1 13 3 5 8 2 9 4 8: 1 11 2 7 5 9: 2 10 5 12 6 11 3 13 4 7 10: 0 13 1 5 9 2 11: 1 3 9 6 2 8 12: 5 6 9 13: 0 2 4 9 3 7 1 10 2. G 2 T r i a n gu l a t i on : 0: 1 12 7 6 4 11 5 3 8 9 1: 0 9 8 12 2: 3 5 12 3: 0 5 2 12 10 4 8 4: 0 6 12 5 8 3 10 11 5: 0 11 13 8 4 12 2 3 6: 0 7 10 8 13 12 4 7: 0 12 11 10 6 8: 0 3 4 5 13 6 10 12 1 9 9: 0 8 1 10: 3 12 8 6 7 11 4 11: 0 4 10 7 12 13 5 12: 0 1 8 10 3 2 5 4 6 13 11 7 13: 5 11 12 6 8 Se v e n t h: D o es the r e exist a g r ap h G 3 2 with ( G 3 2 ) = 20 ? T o ans w er th e se v e n t h q u est i on w e h a v e t o sh o w t h a t t h e up p e r b ound of c hro m a ti c n u m b er for a n y ge n us 2 and t h i c k n es s t h r ee graph i s 20. Then w e apply t h e second t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p os i ti o n 2 and Pr o p os i ti o n 3. Th e Eu l e r c h a r a ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (2) = 2. Th e M p i r e c h r om a t i c n u m b er for a n y su r fa ce wi t h Euler c haracteristic i s at most:

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60 ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (3 pire, 2) (6 3)+1+ (6 3+1) 2 24( 2) 2 20. Th e graph K 20 whi c h has 190 edges i s th e union of t h re e graphs ea c h of whi c h can b e e m b edded on ge n us 2. S i n c e t h e graph t h at has n v er t i ces has a t most 3( n ) edges can b e e m b edded on a su r fac e wi t h Euler c har a ct er i st i c t h en t h e graph K 20 can b e e m b edded on t h e d o ub l e t o r u s ( ge n u s 2) whi c h has 20 v er ti c es has a t most 66 edges. T h u s K 20 i s a graph wi t h ( G 3 2 ) = 20. By second t e c hn i q u e and S u l a n k s T so f t w a re w e found t h e de co m p os i ti o n of G 3 2 = K 20 i n t o t h re e subgraphs ( G 3 2 = G 1 U G 2 U G 3 ) whi c h can b e e m b edded on ge n us 2 as foll o ws: 1. G 1 T r i a n gu l a t i on : 0: 6 14 17 7 19 13 18 11 16 10 8 9 1: 6 9 19 7 11 2: 4 19 18 17 3: 6 17 14 4: 2 17 6 11 19 5: 11 12 17 6: 0 9 1 11 4 17 3 14 7: 0 17 12 9 8 10 16 11 1 19 8: 0 10 7 9 9: 0 8 7 12 15 19 1 6 10: 0 16 7 8 11: 0 18 12 5 17 19 4 6 1 7 16 12: 5 11 18 19 15 9 7 17 13: 0 19 17 18 14: 0 6 3 17 15: 9 12 19 16: 0 11 7 10 17: 0 14 3 6 4 2 18 13 19 11 5 12 7 18: 0 13 17 2 19 12 11 19: 0 7 1 9 15 12 18 2 4 11 17 13 2. G 2 T r i a n gu l a t i on : 0: 1 2 15 1: 0 15 17 16 13 10 18 3 14 2

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61 2: 0 1 14 9 5 7 3 16 12 6 10 15 3: 1 18 16 2 7 5 10 19 8 15 11 14 4: 9 16 18 5: 2 9 18 10 3 7 6: 2 12 10 7: 2 5 3 8: 3 19 12 16 17 15 9: 2 14 16 4 18 5 10: 1 13 15 2 6 12 19 3 5 18 11: 3 15 14 12: 2 16 8 19 10 6 13: 1 16 15 10 14: 1 3 11 15 16 9 2 15: 0 2 10 13 16 14 11 3 8 17 1 16: 1 17 8 12 2 3 18 4 9 14 15 13 17: 1 15 8 16 18: 1 10 5 9 4 16 3 19: 3 10 12 8 3. G 3 T r i a n gu l a t i on : 0: 3 12 5 4 1: 4 8 5 12 2: 8 13 11 3: 0 4 17 9 13 12 4: 0 5 15 7 13 8 1 12 14 10 17 3 5: 0 12 1 8 6 16 19 14 13 15 4 6: 5 8 18 15 13 7 19 16 7: 4 15 18 14 19 6 13 8: 1 4 13 2 11 10 14 18 6 5 9: 3 17 10 11 13 10: 4 14 8 11 9 17 11: 2 13 9 10 8 12: 0 3 13 14 4 1 5 13: 2 8 4 7 6 15 5 14 12 3 9 11 14: 4 12 13 5 19 7 18 8 10 15: 4 5 13 6 18 7 16: 5 6 19 17: 3 4 10 9 18: 6 8 14 7 15 19: 5 16 6 7 14

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62 Ei g h t h: D o es the r e exist a g r a ph G 4 2 with ( G 4 2 ) = 26 ? T o ans w er t h e ei g h t h q u est i o n, w e h a v e t o sh o w t h at t h e up p er b ound of c hromatic n u m b er for a n y ge n us 2 and t h i c k n ess four graph i s 26. Th e n w e apply t h e seco nd t e c hn i q u e, whi c h i s i n sp i r ed b y Pr o p osi t i o n 2 and Pr o p osi t i o n 3. Th e Eu l e r c h a r a ct er i st i c for o ri e n ta b l e su r fa ces g i s ( g ) = 2 2 g so (2) = 2. Th e M p i r e c h r om a t i c n u m b er for a n y su r fa ce wi t h Euler c haracteristic i s at most: ( M pire, ) 6 M +1+ (6 M +1) 2 24 2 Hence, (4 pire, 2) (6 4)+1+ (6 4+1) 2 24( 2) 2 26. Th e graph K 26 whi c h has 325 edges i s th e union of t h re e graphs ea c h of whi c h can b e e m b edded on ge n us 2. S i n c e t h e graph t h at has n v er t i ces has a t most 3( n ) edges can b e e m b edded on a su r fac e wi t h Euler c har a ct er i st i c t h en t h e graph K 26 can b e e m b edded on t h e d o ub l e t o r u s ( ge n u s 2) whi c h has 26 v er ti c es has a t most 84 edges. T h u s K 26 i s a graph wi t h ( G 4 2 ) = 26. By second t e c hn i q u e and S u l a n k s T so f t w a re w e found t h e de co m p os i ti o n of G 4 2 = K 26 i n t o four subgraphs ( G 4 2 = G 1 U G 2 U G 3 U G 4 ) whi c h can b e e m b edded on ge n us 2 as foll o ws: 1. G 1 T r i a n gu l a t i on : 0: 1 20 9 15 24 13 4 3 25 1: 0 25 3 7 19 20 2: 17 20 22 3: 0 4 16 22 20 19 7 1 25 4: 0 13 12 20 14 16 3 5: 13 16 14 6: 13 23 17 7: 1 3 19 8: 9 22 16 21 9: 0 20 13 11 22 8 21 24 17 15 10: 13 14 20 17 24 15 23 11: 9 13 22 12: 4 13 18 20 13: 0 24 16 5 14 10 23 6 17 22 11 9 20 18 12 4 14: 4 20 10 13 5 16 15: 0 9 17 23 10 24 16: 3 4 14 5 13 24 21 8 22

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63 17: 2 22 13 6 23 15 9 24 10 20 18: 12 13 20 19: 1 7 3 20 20: 0 1 19 3 22 2 17 10 14 4 12 18 13 9 21: 8 16 24 9 22: 2 20 3 16 8 9 11 13 17 23: 6 13 10 15 17 24: 0 15 10 17 9 21 16 13 25: 0 3 1 2. G 2 T r i a n gu l a t i on : 0: 2 11 16 10 6 7 22 14 1: 5 6 8 2: 0 14 18 24 22 13 25 7 16 11 3: 6 9 12 24 18 14 4: 5 19 17 7 23 22 15 5: 1 8 12 19 4 15 22 24 7 20 6 6: 0 10 16 9 3 14 8 1 5 20 7 7: 0 6 20 5 24 12 8 14 23 4 17 16 2 25 22 8: 1 6 14 7 12 5 9: 3 6 16 12 10: 0 16 6 11: 0 2 16 12: 3 9 16 19 5 8 7 24 13: 2 22 25 14: 0 22 21 23 7 8 6 3 18 2 15: 4 22 5 16: 0 11 2 7 17 19 12 9 6 10 17: 4 19 16 7 18: 2 14 3 24 19: 4 5 12 16 17 20: 5 7 6 21: 14 22 23 22: 0 7 25 13 2 24 5 15 4 23 21 14 23: 4 7 14 21 22 24: 2 18 3 12 7 5 22 25: 2 13 22 7 3. G 3 T r i a n gu l a t i on : 0: 12 21 18 23

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64 1: 3 13 21 17 2: 4 6 12 23 8 10 3: 1 17 11 10 8 23 5 9 13 4: 2 10 9 18 6 5: 3 23 9 6: 2 4 18 22 19 24 25 15 12 7: 9 10 21 13 8: 2 23 3 10 9: 3 5 23 25 16 18 4 10 7 13 10: 2 8 3 11 19 21 7 9 4 11: 3 17 25 19 10 12: 0 23 2 6 15 25 14 17 21 13: 1 3 9 7 21 14: 12 25 17 15: 6 25 12 16: 9 25 20 23 18 17: 1 21 12 14 25 11 3 18: 0 21 19 22 6 4 9 16 23 19: 6 22 18 21 10 11 25 23 24 20: 16 25 24 23 21: 0 12 17 1 13 7 10 19 18 22: 6 18 19 23: 0 18 16 20 24 19 25 9 5 3 8 2 12 24: 6 19 23 20 25 25: 6 24 20 16 9 23 19 11 17 14 12 15 4. G 4 T r i a n gu l a t i on : 0: 5 17 8 19 1: 2 9 14 24 4 23 11 12 22 10 18 16 15 2: 1 15 3 21 5 19 9 3: 2 15 21 4: 1 24 8 25 21 6 11 23 5: 0 19 2 21 25 10 12 11 18 17 6: 4 21 11 7: 11 15 18 8: 0 17 18 25 4 24 11 20 15 13 19 9: 1 2 19 14 10: 1 22 12 5 25 18 11: 1 23 4 6 21 20 8 24 14 15 7 18 5 12 12: 1 11 5 10 22 13: 8 15 19 14: 1 9 19 15 11 24

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65 15: 1 16 18 7 11 14 19 13 8 20 21 3 2 16: 1 18 15 17: 0 5 18 8 18: 1 10 25 8 17 5 11 7 15 16 19: 0 8 13 15 14 9 2 5 20: 8 11 21 15 21: 2 3 15 20 11 6 4 25 5 22: 1 12 10 23: 1 4 11 24: 1 14 11 8 4 25: 4 8 18 10 5 21 Ni n th: D o es the 10 ch r omatic { 4, 4,4 ,4, 3 } i n fl at e d C 5 have thickness two? In order t o a n s w e r t h e ni n th q u es t i on w e h a v e ap p li e d th e second t e c hn i q u e. In t h i s t e c hn i q u e w e started wi t h { 4 4, 4 4, 3 } i n fl a t ed C 5 on ge n us 0, whi c h i s 10 c h r o m at i c graph with 19 v er t i ces and 99 edges, b y ( De fi n i ti o n 9). Un fo r tu n a t el y w e also h av e not succeeded on finding th a t 10 c hrom a ti c { 4 4, 4 4, 3 } inflated C 5 h a v e t h i c k n ess t w o Th e re w ere 5 edges t h at do n o t b e l on g t o a n y sub graphs. Th e { 4 4, 4 4, 3 } i n fl a t ed C 5 graph has a t most t h i c k n ess th r ee as sh o wn i n figure 6.7. Figure 6.7: D eco m p os i ti o n of { 4 4, 4 4, 3 } i n fl a t ed C 5

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66 T e n th: D o es the 10 ch r omatic 4 inflat e d C 7 have thickness two? In order t o ans w er t h e t e n t h q u est i o n, w e h a v e applied t h e second t e c hn i q u e. In t h i s te c hn i q u e w e st ar t ed wi t h 4 i n fl a t ed C 7 on ge n us 0, whi c h i s 10 c hromatic graph on 28 v er t i ces and 154 edges, b y ( d efi n i t i o n 9). W e also h av e n o t succeeded on finding th a t 1 0 c h r om a t i c 4 i n fl a t ed C 7 h a v e t h i c k ness t w o. Th e re w ere 9 edges t h at do not b elong t o a n y subgraphs. Th e 4 inflated C 7 graph has at most t h i c k n ess t h r ee as sh o wn in fi g u r e 6.8. Figure 6.8: D eco m p osi ti o n of 4 inflated C 7 El e v e n t h: I s the r e a re ctangle visibility r ep r esentation for K 23 in 3 dimen sions? In o rd e r t o a n s w e r t h e el e v e n t h question, w e h a v e applied t h e t h i rd t e c hn i q u e. Th e sea r c h for a r ep r ese n t a ti o n of K 23 ran on cluster computer for a l m os t t h re e m o n th s w i th o u t success. Wh il e a r ect a ngl e v i si b ili t y r ep r ese n ta t i o n for K 22 w a s found in appr o ximately one w ee k. In Ap p e nd i x A, w e su mm a r i zed t h e b est found r esu l t s of applying first t e c hn i qu e wi t h A CO, F A and SA al go r i th m s in order to ans w er first, second and third q u es t i on s

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67 6.2 Heur is t ic Meth o ds Co mpa ri s on Th i s study aimed a t applying and co mpar i n g t h e d i ffer e n t heuristic meth o d s on so l v i n g coloring p r ob l e m s in graphs and vi s i b ili t y r ep r ese n ta t i on for graphs i n 3 d i m en si o n s. W e h av e used four d i ffer e n t heuristic m e th o d s t h r ou g h o u t t h r ee t e c hn i q u es ( see c hapter 5). Th e a l go r i th m s w ere e v a l u at ed u s i n g t w o cr i t er i a: t h e q ual i t y of th e s olu t i o n s and t h e r at e of t i m e s p en d i n g to find an o p t i mal so l u t i on In first t e c hn i qu e w e i mp l em e n ted t h r ee d i ffer e n t op t i m i za t i on a l go r i t hm s ( A n t Colo n y Op t i m i zat i o n ( A CO), Firefly Al g or i t hm ( F A), and S i m u l a ted Annealing ( S A) ). W e h a v e c hosen A CO and F A due t o t h e fact t h at d i ffer e n t a n t s and d i f fer e n t fi re fl i es will w ork almost i nd e p e nd e n t l y p a r ti c u l ar l y t h a t t h ey are parallel i mp l em e n t a t i on ; wh il e SA has b een c hosen b ecause it w as th e al go ri t hm t h a t used t o find 9 cr i t i ca l graphs wi t h t h e same te c hn i q u e in [GS09]. A fu r t h er ad v a n t ag e of A CO and F A i s t h a t t h ey are e v en b etter t h an SA b ecause a n t s and fi r efl i es agg r ega t e more closely around ea c h op t i m um so l u t i on wi t h ou t jumping around a s in t h e case of SA. All these al go ri t hm s pr o duced a l m os t t h e same q u a li t y of g o o d so l u t i on H ow e v er, t h e d i ffe re n ce i s in t h e a m ou n t of t i m e t h a t ea c h algorithm s p e n t t o g i v e t h e so l u t i on W e can see h o w t h e s w arm i n te lli gen c e al go ri t hm s o u t p erformed t h e si m u l at ed a nn ea li n g in t h e foll o wing figures for finding: or i e n t a b l e ge n us 0 9 c ri t i ca l t h i c k n es s t w o graphs on 17 v er t i ces whose com p l e m e n t s a r e K 3 free ( fi gu r e 6.9), o ri e n ta b l e ge n us 0 9 cr i t i ca l t h i c k n ess t w o graphs on 25 v er t i ces whose c om p l e m e n t s are K 4 f re e ( fi gu r e 6.10), or i e n t a b l e ge n us 0 9 cr i t i ca l t h i c k ness t w o graphs on 33 v er ti c es whose co mp l em e n t s are K 5 f re e (fi g u r e 6.11), wh er e x a x i s i s t h e i n i t i a l seeds and y a x i s i s t h e t i m e on sec on d s. Figure 6.9: First T e c hn i q u e Al go ri t hm s P erformance ( 1 )

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68 Figure 6.10: F i rs t T e c hn i q u e Al g or i t hm s P erformance ( 2) Figure 6.11: F i rs t T e c hn i q u e Al g or i t hm s P erformance ( 3) S i m u l a ted an n e al i n g a l w a y s accep t s a selected b et t er co st l o cal sol u t i o n and i t m a y also accep t a w o r se c ost l o cal s olu t i o n wi t h a prob a b ili t y whi c h i s gradually decreased in a l go r i th m s e xe cu t i on As a r esu l t it p er fo r m s w orse than s w arm i n t el li gen ce al go ri t hm s since t h e last only accep t t h e b e t te r co st l o ca l so l u t i on s Ou r r esu l t s h av e sh o wn t h at t h e A CO i s su p er i o r to F A in t er m s of t i m e. More o v er, t h e r esu l t s sh o w t h a t A CO p r es e n t ed t h e b est b a l an ce b e tw een t h e t w o cr i t er i a.

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69 Th e third t e c hn i qu e t h at w e h a v e i mp l e m e n t ed b y t h e t h r ee d i ffe re n t S w arm I n t el li gen ce ( S I) meth o d s ( A n t Colo n y Op t i m i za t i o n ( A CO), Cu ck o o Op ti m i z at i o n Al g or i t hm (C O A), and Firefly Al g or i t hm ( F A)). T h e main c har a ct er i st i c fe at u r e of t h e S I a l go r i t hm s i s t h e fact t h at t h ey si m u l a te a p a r al l el i nd e p en d e n t run st ra t eg y where in e v ery i t er at i o n, a s w arm of n ( a n t s, cu c k o os, and fi r efl i es ) h av e ge n er at ed n so l u t i on s. Ea c h (a n t, cu c k o o, and firefly) w orks almost i nd e p e nd e n t l y and as a r esu l t t h e a l go r i t hm will find t h e o p t i mal s olu t i o n v ery qui c kl y F i gu r e 6.12 s h o ws t h e p erformance of running t h e a l go r i th m s in t h i rd t e c hn i q u e t o find vi s i b ili t y r ep r ese n t a t i on for co mp l et e graphs K n b y r ect an g l es in 3 d i m e n si on s Figure 6.12: Third T e c hn i q u e Al g or i t hm s P e rf orm a n ce As b efore, t h e A CO o u t p er fo rm e d F A, bu t C O A i s b et t er t h a n A CO. Th e m a jor d i sa d v a n t a ge in t h e A CO i s t h at wh il e t r y i n g to sol v e t h e op t i m i za t i on p r ob l em s t h e a n ts will w alk t h r ou g h t h e path where t h e pheromone has b een d e p os i ted, wh il e t h e l o cal sear c h has t o p er fo rm e d m u c h faster. Hence, l o cal sea r c h will b e p erforming a t t h e faster rate in C O A than in t h e A CO, since t h e m o v em e n t of cu c k o o in C O A i s a random w alk. Mo r e o v e r, t h e main a d v a n t ag e of C O A i s t h a t it can sol v e o p ti m i z at i o n p r o b l em s based on t h a t in e v ery i tera t i o n, a fraction of w o r se so l u t i on s are abandoned and in e v ery gen e ra t i on cu c k o os a r e m o ving t o w a r d s an op t i m i ze d sol u t i o n b y r ep l a cin g t h e sol u t i o n s wi t h t h e g o o d ones. By t h e end of maxi m um i tera t i on s w e will o b t ain t h e o p t i m i zed so l u t i on O v e ra ll t h e r esu l t s i nd i c at e th a t all four meth o d s can b e used for so l vi n g com binatorial p r ob l e m s. Th e st r eng t h of t h e t h re e S I a l go r i th m s i s in p a r al l e l wi t h t h e sea r c h approa c h t h ey e mp l o y In co mpu t at i o n t er m s, C O A approa c h i s faster,

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70 although none of t h e S I a l go r i t hm s t a k e a long ti m e li k e SA al go ri t hm. Th e re su l t s for SA suggest th a t wh il e it i s a v a l u a b l e te c hn i q u e, it d o es n o t p r o v i d e a n y i mp r o v em e n t o v er t h e other t h r ee t e c hn i q u es. Th e decision on wh e th e r to c h o o se b e tw een C O A or A CO m e th o d i s related t o t h e e v a l u a ti o n c ri t er i a : th e q ual i t y of t h e so l u t i on s suggest b oth A CO and C O A, wh il e t h e fast rate t i m e can b e o b t ain ed b y C O A. 6.3 F uture W o rk Th e r esu l t s of t h i s t h es i s p o i n t t o se v er a l i n t er es ti n g d i r ec ti o n s for fu t u r e w ork: One su c h direction w ould b e t o consider some co m b i n at i o n s b e tw een A CO and C O A a l go r i t hm s as an a l t er n at i v e approa c h to sol v e these and o t h er p r ob l e m s e v en more effi ci e n t l y A p o ssi b l e approa c h for t h i s i s t o form a pi p eline, p as si n g t h e r esu l t s from one a l go r i t hm t o a n o th e r or in p a r al l e l c h o osing t h e b est r es u l t from t h e parallel ru n s of se v er a l a l go r i th m s or b y h a ving co m m un i ca ti o n b e tw e en t h e a l go r i th m s Therefore, w e b el i e v e t h a t h ybrid or c o o p er a t i v e op t i m i za t i on st r at eg i es for c o m b i n a t or i a l graph t h eo ry p r ob l e m s a r e a p r om i s i n g fu t u r e re sea r c h. An o th e r p os si b ili t y w ould b e t o d es i gn an ex h au st i v e sear c h for all d eco m p os i ti o n s of K 14 i n t o t w o subgraphs whi c h can b e e m b edded on th e d ou b l e t oru s, K 20 i n t o t h r ee subgraphs whi c h can b e e m b edded on t h e d ou b l e t oru s, and K 26 i n t o four subgraphs whi c h can b e e m b edded on t h e double torus.

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71 APPENDIX A EXPERIMENT R ESU L TS Th e foll o wing t ab l es su mm a ri z ed t h e b est found r esu l t s of applying first t e c hn i q u e wi t h A CO, F A and SA al go r i th m s in order to ans w er first, second and third q u es t i on s ( see r ese ar c h q u es ti o n s) b y using CCM cluster co mpu t er In T a b l e A.1 w e sh o w ed t h e re su l t s of t h e first t e c hn i q u e b y A CO al go ri t hm. T a b l e A.2 sh o ws t h e r esu l t s of t h e fi rs t te c hn i q u e b y F A a l go r i t hm, wh il e T a b l e A.3 sh o ws t h e r esu l t s of t h e first t e c hn i q u e b y SA algorithm. T a b l e A.1 : Th e R esu l t s of First T e c hn i q u e b y A CO Al g or i t hm I ni ti al s eed V K m f re e Num. of K m s ta r t w it h Min Num. o f K m 100 2400 60000 19 28 37 10 10 10 2 2 2 K 3 K 4 K 5 23 265 312 5 71 43 700 2000 90000 21 31 41 11 11 11 2 2 2 K 3 K 4 K 5 64 934 715 35 169 140 1100 3000 12000 23 34 45 12 12 12 2 2 2 K 3 K 4 K 5 133 1262 901 95 69 162 800 1600 80000 33 49 65 17 17 17 3 3 3 K 3 K 4 K 5 420 1343 1095 307 332 696 1400 2000 19000 35 52 69 18 18 18 3 3 3 K 3 K 4 K 5 559 1506 1170 371 606 754 500 3000 60000 45 67 89 23 23 23 4 4 4 K 3 K 4 K 5 790 1808 2200 199 359 459 100 4000 19000 47 70 93 24 24 24 4 4 4 K 3 K 4 K 5 823 2010 3511 236 352 881

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72 T a b l e A.2 : Th e R esu l t s of First T e c hn i q u e b y F A Al g or i t hm I ni ti al s eed V K m f re e Num. of K m s ta r t w it h Min Num. o f K m 1300 2200 14000 19 28 37 10 10 10 2 2 2 K 3 K 4 K 5 20 284 410 5 80 122 200 7000 50000 21 31 41 11 11 11 2 2 2 K 3 K 4 K 5 69 945 765 38 106 144 500 1000 61000 23 34 45 12 12 12 2 2 2 K 3 K 4 K 5 105 1227 886 93 115 162 200 7000 31000 33 49 65 17 17 17 3 3 3 K 3 K 4 K 5 423 1360 1172 307 335 696 800 5000 18000 35 52 69 18 18 18 3 3 3 K 3 K 4 K 5 580 1475 1222 371 607 755 1100 3300 40000 45 67 89 23 23 23 4 4 4 K 3 K 4 K 5 799 1820 2311 198 359 460 1600 6000 100000 47 70 93 24 24 24 4 4 4 K 3 K 4 K 5 816 1999 3312 236 351 880

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73 T a b l e A.3 : Th e R esu l t s of First T e c hn i q u e b y SA Al g or i t hm I ni ti al s eed V K m f re e Num. of K m s ta r t w it h Min Num. o f K m 500 2000 31000 19 28 37 10 10 10 2 2 2 K 3 K 4 K 5 16 299 50 5 71 45 1400 5000 100000 21 31 41 11 11 11 2 2 2 K 3 K 4 K 5 70 832 690 37 109 142 800 2300 70000 23 34 45 12 12 12 2 2 2 K 3 K 4 K 5 126 1409 891 97 229 162 1400 2500 30000 33 49 65 17 17 17 3 3 3 K 3 K 4 K 5 441 1402 1210 307 390 699 900 2800 30000 35 52 69 18 18 18 3 3 3 K 3 K 4 K 5 512 1380 1078 371 606 771 500 1500 70000 45 67 89 23 23 23 4 4 4 K 3 K 4 K 5 654 1795 1982 198 359 460 1600 4000 20000 47 70 93 24 24 24 4 4 4 K 3 K 4 K 5 791 1989 2155 237 351 883

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74 APPENDIX B NEW N I NE CRI T I CA L GRA PH S W e will i n t r o du ce thir t y 9 cr i t i cal th i c kn e ss t w o graphs on 17 v er t i ces wi t h K 3 free co mp l em e n t s, thir t y 9 cr i t i ca l t h i c k n ess t w o graphs on 25 v er t i ces wi t h K 4 f re e co mp l em e n t s and thir t y 9 c ri t i ca l t h i c k n ess t w o graphs on 33 v er ti c es wi t h K 5 f re e co mp l em e n t s b y using ( A CO, F A, and S A) al g or i t hm s. Th e graphs w ere c h ose n for t h e un i q u en ess of degree sequences. T o t h at end, ea c h graph G i s i d e n t i fi e d b y i t s d eg r ee seq u en c e. In o rd e r t o d i s p l a y a t h i c k n ess t w o e m b ed d i n g for ea c h graph w e used Gr o up s and Gr a ph s [M c k12]. T h e final g r ap h i c s w ere generated b y Ma t h em a ti c a [ W ol12]. Th e 90 new 9 cr i ti c al t h i c k n es s t w o graphs foll o w. B.1 Thi c k ne ss tw o graphs on 17 v ertices whose com pleme n ts are K 3 f re e B.1.1 A n t Colo n y Algorithm ( A CO )

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75 Degree Sequence {1 3 1 2 1 2 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9} Degree Sequence { 1 3 1 2 11 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9}

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76 D eg r ee S e q u e n ce { 1 3 1 2 11 11 1 1 1 1 11 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9} Degree S e q u ence { 11 11 11 11 11 11 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9}

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77 Degree Sequence {1 2 1 2 1 1 11, 1 1 11 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9} Degree Sequence { 1 3 1 2 11 11 11 1 1 1 1 11 1 0 1 0 1 0 1 0 9 9 9 9 9}

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78 D e gree Sequence {1 3 1 2 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9} Degree Sequence { 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9}

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79 B.1.2 Firefly Algorithm ( F A)

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80 D e gree Sequence { 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9} Degree Sequence { 1 3 1 2 1 1 11 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9}

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81 Degree Sequence {1 2 1 2 1 2 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9} Degree Sequence { 1 3 1 2 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 }

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82 Degree Sequence { 1 3 1 1 11, 1 1 11 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9} Degree Sequence { 1 2 1 1 11, 11, 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 }

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83 D eg r ee S e q u e n ce { 1 3 1 3 1 3 1 1 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9} Degree S eq u ence { 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9}

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84 B.1.3 Si m ula t ed Annealing Algorithm ( SA)

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85 Degree Sequence { 1 3 1 2 11 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9} Degree Sequence { 1 2 1 2 1 2 1 1 11 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9}

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86 D e gree Seque n ce { 1 2 1 2 1 2 1 1 1 1 11 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9} D e g r e e Seq u e n ce { 1 3 1 2 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9}

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87 Degree Sequence { 1 3 1 2 1 1 1 1 11 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9} Degree Sequence { 1 2 1 2 11 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9}

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88 D egree S e q u e n ce {12 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9} Degree S e q u ence { 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 }

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89 D eg r ee Seque n ce { 1 2 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9}

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90 B.2 Thi c k ne ss tw o graphs on 25 v ertices whose com pleme n ts are K 4 f re e B.2.1 A n t Colo n y Algorithm ( A CO )

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91 D e gree Sequence { 1 3 1 3 1 1 1 1 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 6 1 5 1 3 1 2 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8}

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92 Degree Sequence {1 4 1 4 1 3 13, 11 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Sequence { 1 4 1 2 1 2 11 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8}

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93 Degree Sequence {1 3 1 3 1 2 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 5 1 2 1 2 11 1 1 11, 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 8 8 8 8 8 8 8}

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94 Degree Sequence {1 3 1 2 1 2 1 1 11 1 1 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Sequence { 1 6 1 2 1 2 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 8 8 8 8 8 8 8}

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95 B.2.2 Firefly Algorithm ( FA ) Degree Sequence { 1 4 1 3 1 2 1 2 1 2 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Sequence { 1 3 1 2 1 2 1 1 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8}

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96 Degree Sequence { 1 3 1 3 1 2 1 2 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 7 1 2 1 2 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9}

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97 Degree Sequence {1 3 1 3 1 2 1 1 11 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 4 1 2 1 2 1 1 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8}

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98 Degree Sequence {1 4 1 1 1 1 11 11, 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 4 1 3 1 3 1 3 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9}

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99 Degree Sequence {1 3 1 3 1 2 1 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 8 8 8 8 8 8} Degree Sequence { 1 3 1 2 1 2 1 2 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8}

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100 B. 2 3 Simulated Annealing Algorithm (SA) Degree Sequence { 1 4 1 3 1 2 1 1 1 1 1 1 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Sequence { 1 5 1 2 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8}

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101 Degree Sequence { 1 7 1 2 1 2 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 4 1 3 1 3 1 2 1 2 1 1 9 9 9 9 9 9 9 9 9 9 9, 9 9 9 9 9 9 9 9}

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102 D e gree Sequence { 1 3 1 2 1 2 1 1 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 3 1 2 1 2 1 1 11 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 8 8 8 8 8 8 8}

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103 Degree Sequence {1 3 1 3 1 2 1 1 11 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8} Degree Sequence { 1 5 1 3 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9}

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104 Degree Sequence { 1 4 1 3 1 2 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 8 8 8 8 8} Degree Sequence { 1 6 1 2 11 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8}

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105 B.3 Thi c k ne ss tw o graphs on 33 v ertices whose com pleme n ts are K 5 f re e B.3.1 A n t Colo n y Algorithm ( A CO )

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106 D e g r e e Sequence { 1 4 1 4 1 3 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8} D e gree Sequence { 1 5 1 4 1 4 1 3 1 2 1 2 1 2 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9, 9 9 9 9, 8 8 8 8 8 8}

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107 Degree Se q u ence { 1 4 1 3 1 3 1 3 1 2 1 2 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Se q u ence { 1 3 1 1 1 1 1 1 1 0 1 0 1 0, 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8}

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108 Degree Sequ e n ce { 1 6 1 2 1 2 1 2 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8} Degree Sequence {1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9}

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109 Degree Sequence {1 4 1 4 1 3 1 1 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8} Degree Sequence {1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8}

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110 B.3.2 Firefly Algorithm ( F A)

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111 De g ree Se q u e n ce { 1 5 1 4 1 2 1 L 1 1 1 1 1 1 1 1 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8} De g r ee Sequ e n ce { 1 7 1 3 1 2 1 2 1 1 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8}

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112 Degree Se q u e n ce { 1 2 1 2 12 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8} Degree S e q u e n ce { 1 5 1 4 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8}

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113 Degree Sequence {1 3 1 3 1 1 1 1 11 11 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8} Degree Sequence { 1 4 1 3 13, 1 2 1 2 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8}

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114 Degree Sequence { 1 5 1 3 1 2 1 2 1 1 1 1 1 1 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8} Degree Se q u ence { 1 2 1 2 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8}

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115 B.3.3 Si m ula t ed Annealing Algorithm ( SA)

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116 Degree Sequen c e { 1 5 1 3 1 2 1 1 1 1 1 1 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 8 8} D e gree Sequence { 1 2 1 2 1 1 1 1 1 1 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8}

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117 Degree Se q u e n ce { 1 2 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8} Degree Sequence { 1 2 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9, 9}

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118 Degree Sequence { 1 4 1 3 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9} Degree Sequ e n ce { 1 2 1 2 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8}

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119 D e g r ee Se q u e n ce {1 3 1 2 1 1 1 1 1 1 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8} De g r ee Se q u e n c e { 1 2 1 1 1 1 1L 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8}

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120 Degree Se q u ence { 1 5 1 3, 1 1 11 11 11 11 1 0 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8, 8 8 8 8 8 8 8}

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121 B I BL I OG RAP HY [ABG10] M i c h ae l O. Al b er t son Debra L. Bo u t i n, and Ellen Gethner. T h e t h i c k ness and c h r om a t i c n u m b er of r i n fl a t ed graphs. Disc r ete Mathe matics 310 ( 20 ) : 2725 2734 2010. 5, 12 [ABG11] Michael O. Albertson, Debra L. Boutin, and Ellen Gethner. More results on r inflated graphs: arboricity, thickness, chromatic number and fractional chromatic number. Ars Math. Contemp., 4(1):5 24, 2011. 12 [AH74] Alfred V. Aho and John E. Hopcroft. The Design and Analysis of Computer Algorithms. Addison Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition, 1974. 6 [AH77] K. Ap p el and W. Ha k e n. E v ery planar map i s four c ol o r ab l e p a r t I: D i s c h a r gin g I l linois Journal of Mathematics 21:429 490, 1977. 6, 14, 20 [AH88] Mi c hael O. Al b ertson and Joan P Hu t c h i n so n. Disc r ete mathemat ics with algorithms John Wiley & Sons, Inc., New Y ork, NY, USA, 1988. 6, 8, 9, 10 [AMS96] Isto Aho, Erkki M ¨ a ki n e n, and T arja S y st ¨ a R em a rk s on t h e t h i c k n ess of a graph. T e c hn i ca l Re p ort A 1996 2, D epa rt m e n t of Co mpu t er Science, Un i v er si t y of T am p ere, 1996. A revised v e rs i on ap p eared i n Information Scineces 108 (1998), 1 4. 13, 14 [BDHS97] Pr os en ji t Bose, Alice Dean, Joan Hu t c h i n so n, and Thomas S h er m er On re ct an g l e v i si b ili t y graphs, 1997. 5, 18, 24, 26 [ BE F + 9 4] P Bose, H. E v er et t S. P F e k ete, A. Lubiw, H. Mei j er K. Romanik, T. Shermer, and S. Wh i t esi d e s. On a vi s i b ili t y r ep r ese n t a t i on for graphs in t h r ee d i m en s i on s T e c hn i ca l r e p o r t Ma t h em a ti s c h es In st i tut, Un i v e rs i t ¨ a t zu K ¨ o l n, 1994. 2, 25, 27, 28, 48 [Bei97] Beine k e. Biplanar graphs: A su r v e y CANDM: A n I nter national Journal: Computers and Mathematics, with Appli c ations 34, 1997. 18, 22, 24 [ BE L + 9 3] P ro sen j i t Bose, Hazel E v e re tt Anna Lubiw, Henk Meijer, Kathleen Romanik, Th o m a s C. Shermer, Sue Wh i t esi d e s, Ch ri s ti a n Zel l e, S a n dor F e k ete, M i c h ae l E. Houle, and G A n t er Ro t e. A v i si b ili t y repre se n t a t i on for graphs in t h re e dimensions. In J. GRAPH A L GO RI TH M S APPL pages 2 25, 1993. 5, 24, 25, 26, 48

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