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Incidence Labeling Games on Incidence-Symmetric Graphs (2,∞)-INCIDENCE LABELING GAMES ON INCIDENCE-SYMMETRIC GRAPHS BY MS. NATTHAWAN SRIPHONG A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (MATHEMATICS) DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE AND TECHNOLOGY THAMMASAT UNIVERSITY ACADEMIC YEAR 2018 COPYRIGHT OF THAMMASAT UNIVERSITY Ref. code: 25616009031029DVT (2,∞)-INCIDENCE LABELING GAMES ON INCIDENCE-SYMMETRIC GRAPHS BY MS. NATTHAWAN SRIPHONG A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (MATHEMATICS) DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE AND TECHNOLOGY THAMMASAT UNIVERSITY ACADEMIC YEAR 2018 COPYRIGHT OF THAMMASAT UNIVERSITY Ref. code: 25616009031029DVT (1) Thesis Title (2,∞)-INCIDENCE LABELING GAMES ON INCIDENCE-SYMMETRIC GRAPHS Author Ms. Natthawan Sriphong Degree Master of Science (Mathematics) Department/Faculty/University Mathematics and Statistics Faculty of Science and Technology Thammasat University Thesis Advisor Nantapath Trakultraipruk, Ph.D. Academic Year 2018 ABSTRACT A(p,q)−incidence labeling game is a game of coloring on incidences of a graph G. There are two players, Alice and Bob (Alice begins). They alternately label an uncolored incidence of G with a color taken from a set of p colors, and each color can be used at most q times. Adjacent incidences must be labeled by different colors. We determine a winning strategy of a (2,∞)−incidence labeling game on an incidence-symmetric graph. Keywords : incidence labeling game, coloring game, incidence-symmetric graph Ref. code: 25616009031029DVT (2) ACKNOWLEDGEMENTS First of all, I would like to specially thank my wonderful thesis advisor, Dr.Nantapath Trakultraipruk, for his direction, gentleness in giving advice, and support throughout my research. This helped me to complete this thesis. I am also deeply thankful to my thesis committees, Associate Professor Dr.Wichai Witayakiattilerd, Assistant Professor Dr.Khajee Jantarakhajorn, and Assistant Professor Dr.Chanon Promsakon, who gave valuable comments. I also recognize the financial support of the Science Achievement Scholarship of Thailand. In addition, I would like to thank all the staff members from the Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, and all of my friends for their friendship. Finally, I would like to thank my parents for their love and support. This achievement would not have been possible without them. Natthawan Sriphong Ref. code: 25616009031029DVT (3) LIST OF FIGURES Figures Page 1.1 The graph G1 .............................. 1 1.2 Pairs of adjacent incidences . 2 1.3 Pairs of nonadjacent incidences . 2 1.4 Graphs G2 and G3 ........................... 2 1.5 A path P4 ................................ 2 1.6 A cycle C5 ................................ 3 1.7 A complete graph K6 .......................... 3 1.8 A bipartite graph G4 .......................... 3 1.9 The ladder graph L4 .......................... 4 1.10 The crown graph C4,4 ......................... 4 1.11 The friendship graph F4 ........................ 5 3.1 The incidence-symmetric graph G5 .................. 7 3.2 The non incidence-symmetric graph G6 ................ 8 3.3 The path Pn ............................... 9 3.4 The cycle Cn .............................. 11 3.5 The complete graph Kn ........................ 17 3.6 The ladder graph Ln .......................... 22 3.7 The crown graph Cn,n ......................... 26 3.8 The friendship graph Fn ........................ 29 Ref. code: 25616009031029DVT (4) TABLE OF CONTENTS Page ABSTRACT (1) ACKNOWLEDGEMENTS (2) LIST OF FIGURES (3) CHAPTER 1 INTRODUCTION 1 1.1 Basic Knowledge in Graph Theory . 1 1.2 Basic Knowledge in Game Theory . 5 CHAPTER 2 LITERATURE REVIEW 6 CHAPTER 3 (2,∞)-INCIDENCE LABELING GAMES 7 CHAPTER 4 CONCLUSION 36 REFERENCES 37 BIOGRAPHY 38 Ref. code: 25616009031029DVT 1 CHAPTER 1 INTRODUCTION In this chapter, we give some basic knowledge in graph theory and in game theory, which are used in this thesis. For notations and terminology in graph theory, we in general follow [4] and [7]. 1.1 Basic Knowledge in Graph Theory A graph G is a triple, which consists of a vertex set V (G), an edge set E(G), and a relation associating with each edge and two vertices (not necessarily different) called the endpoints. If an edge e has endpoints u and v, we write e = uv (the order does not count). Then u and v are said to be adjacent, and u and e are said to be incident. For example, let G1 be the graph shown in Figure 1.1. Then V (G1) = {v1,v2,v3,v4,v5} and E(G1) = {v1v2,v1v5,v2v3,v2v4,v3v4,v4v5}. v1 v3 v2 G1 : v5 v4 Figure 1.1: The graph G1 Let G be a graph. An incidence of G is defined as a pair (v,e), where v ∈ V (G) is incident to e ∈ E(G). Let I(G) denote the set of all incidences in G. Two distinct incidences (v,e) and (u,f) are adjacent in G if one of the following conditions holds: 1. v = u and e 6= f; 2. e = f and v 6= u; and 3. e = vu, f = uw, and v 6= w. For example, some pairs of adjacent incidences are shown in Figure 1.2, and Ref. code: 25616009031029DVT 2 some pairs of nonadjacent incidences are shown in Figure 1.3. Note that a white dot denotes a vertex, and a black dot on an edge denotes an incidence. Figure 1.2: Pairs of adjacent incidences Figure 1.3: Pairs of nonadjacent incidences A loop is an edge with the same endpoints. Multiple edges are edges, whose pair of endpoints are the same. A simple graph is a graph containing no loops or multiple edges. For example, in Figure 1.4, the graph G2 is a simple graph, but G3 is not a simple graph since it contains multiple edges. G2 : G3 : Figure 1.4: Graphs G2 and G3 A path is an alternating sequence of vertices and edges such that all vertices (except perhaps the first and the last vertices) are distinct. A path with n vertices is denoted by Pn. For example, a path P4 is shown in Figure 1.5. P4 : Figure 1.5: A path P4 Ref. code: 25616009031029DVT 3 A cycle is a path such that the first and the last vertices are the same. A cycle with n vertices is denoted by Cn. For example, a cycle C5 is shown in Figure 1.6. C5 : Figure 1.6: A cycle C5 A complete graph is a simple graph whose vertices are pairwise adjacent. A complete graph with n vertices is denoted by Kn. For example, a complete graph K6 is shown in Figure 1.7. K6 : Figure 1.7: A complete graph K6 A graph G is bipartite if we can partition V (G) into two subsets U and W , such that each edge of G joins a vertex in U and a vertex in W . For example, a bipartite graph G4 is shown in Figure 1.8. G4 : Figure 1.8: A bipartite graph G4 Ref. code: 25616009031029DVT 4 The ladder graph with 2n vertices is denoted by Ln. The vertex set of Ln is V (Ln) = U ∪ V , where U = {u1,u2,...,un} and V = {v1,v2,...,vn}. The edge set of Ln is E(Ln) = C ∪ D ∪ E, where C = {ci = uiui+1 : i ∈ {1,2,...,n − 1}}, D = {di = vivi+1 : i ∈ {1,2,...,n − 1}}, and E = {ei = uivi : i ∈ {1,2,...,n}}. For example, the ladder graph L4 is shown in Figure 1.9. e1 u1 v1 c1 d1 e2 u2 v2 L4 : c2 d2 e3 u3 v3 c3 d3 e4 u4 v4 Figure 1.9: The ladder graph L4 The crown graph with 2n vertices, denoted by Cn,n, is a bipartite graph with partite sets U = {u1,u2,...,un} and V = {v1,v2,...,vn}. The edge set is E(Cn,n) = {ei,j = uivj : i,j ∈ {1,2,...,n} and i 6= j}. For example, the crown graph C4,4 is shown in Figure 1.10. u1 v1 u2 v2 C4,4 : u3 v3 u4 v4 Figure 1.10: The crown graph C4,4 The friendship graph with 2n+1 vertices, denoted by Fn, is the graph with vertex set V (Fn) = {v0,v1,...,v2n} and edge set E(Fn) = E ∪ F , where E = {ej = v0vj : j ∈ {1,2,...,2n}} and F = {fk = vkvk+1 : k ∈ {1,3,...,2n−1}}. For example, the friendship graph F4 is shown in Figure 1.11. Ref. code: 25616009031029DVT 5 v1 f1 v2 v8 e1 e2 v3 e8 e3 F4 : f7 e e f3 7 v0 4 v7 e6 e5 v4 v6 f5 v5 Figure 1.11: The friendship graph F4 1.2 Basic Knowledge in Game Theory For notations and terminology in game theory, we in general follow [3] and [6]. Game theory is the logical analysis of situations of conflict and cooperation. A player is an agent who makes decisions in a game. A strategy is one of the possible actions of a player. A game is a situation which has at least two players, and each player has some possible strategies. A game is finite if these are a finite number of players and moves that a players can make. A game has perfect information if all players are aware of all aspects of the game until the game is over. Chess and tic-tac-toe are examples of perfect information games. Card games such as bridge and poker where each player’s cards are hidden are examples of imperfect information games. Theorem 1.1. [8] In any finite two-person game of perfect information, if the game cannot end in a draw, then one of the players must have a winning strategy. Ref. code: 25616009031029DVT 6 CHAPTER 2 LITERATURE REVIEW A graph coloring game was introduced by Bodlaender [2].
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