Asymmetric Game Perfect Graphs and the Circular Coloring Game of Weighted Graphs Dissertation zur Erlangung des Doktorgrades an der Fakulta¨t fu¨r Mathematik der Universita¨t Bielefeld vorgelegt von Panagiotis Zacharopoulos Oktober, 2011 ii Erster Gutachter: Prof. Dr. Michael Baake Zweiter Gutachter: Prof. Dr. Eckhard Steffen Contents 1 Preliminaries 1 2 Asymmetric Game Perfect Graphs 5 2.1 Asymmetric Game Perfect Graphs . 7 2.2 A(1;n)-Game Perfect Graphs with Clique Number 2 . 8 2.3 B(1;n)-Game Perfect Graphs with Clique Number 2 . 11 2.4 B(n;1)-Game Perfect Graphs with Clique Number 2 . 12 2.5 Construction of B(n;1)-Nice Graphs with Clique Number 2 . 23 2.6 A(n;1)-Game Perfect Graphs with Clique Number 2 . 24 2.7 (n; 1)-Game Perfect Graphs with Clique Number 2 . 32 2.8 (1; n)-Game Perfect Graphs with Clique Number 2 . 34 2.9 (n; m)-Game Perfect Graphs with Clique Number 2 and n; m ≥ 2 34 3 The Circular Two-Person Game on Weighted Graphs 37 3.1 The Circular Two-Person Game on Weighted Graphs . 39 3.2 The Circular Game Chromatic Number of Weighted Complete Graphs . 43 3.3 The Circular Game Chromatic Number of Weighted Complete Multipartite Graphs . 54 3.4 The Circular Game Chromatic Number of Weighted Cycles . 78 3.5 The Circular Game Chromatic Number of Weighted Trees . 83 iii iv CONTENTS 3.6 The Circular Game Chromatic Number of Weighted Planar Graphs . 86 3.6.1 The Circular Marking Game on Weighted Graphs . 86 3.7 The Circular Game Chromatic Number of Weighted Cartesian Product Graphs . 92 4 Circular Game-Perfect Graphs 103 4.1 Circular Game-Perfect Graphs . 104 2 4.2 Circular Game-Perfect Graphs with Θ(G) = K5 . 105 Bibliography 113 Acknowledgments First, I would like to thank my supervisor Prof. Dr. Hans-Georg Carstens for giving me the opportunity to write this thesis. I am very grateful to him for his helpful advice, his motivating comments and permanent support throughout this period. I would like to express my sincere gratitude to Anush Khacha- tryan for so many fruitful discussions, for her patience and constant motiva- tion. Further I would like to thank Prof. Dr. Michael Baake for taking the part of the first supervisor after Prof. Dr. Carstens' death1. Last but not least, I am very thankful to my parents who supported me all the time. 1y28.01.2012 v vi CONTENTS Introduction A graph G is defined to be a pair of vertices and edges. The vertices are denoted V by V and the edges by E ⊆ 2 . Two vertices x; y 2 V are called adjacent if (x; y) 2 E. One of the most inter esting and important branches in the theory of graphs is the theory of graph coloring. A vertex-coloring of a graph G = (V; E) is a map c : V ! S such that c(a) =6 c(b) whenever a and b are adjacent. The elements of the set S = f1; :::; kg are called the available colors. The chromatic number of a graph G, denoted by χ(G), is the minimum integer k such that G is colorable. Obviously the chromatic number is an element of N. Graph coloring has found a lot of applications in daily life; it can be used for instance to optimize time schedules or assignment problems. 1991 H. L. Bodlaender raised the question of how many colors are required for a feasible coloring while considering the vertex coloring of a graph G as a game: two players Alice and Bob take turns coloring the vertices of a given graph G from a prespecified set of colors with Alice starting the game. Alice wins the game if the result is a feasible coloring and Bob wins the game if there exists at least one uncolored vertex and a feasible coloring is not possible any more. In [12] H. L. Bodlaender introduced the coloring game on a graph G and defined the game chromatic number, denoted by γ(G), as the least integer k such that there exists a winning strategy for Alice for a given number of colors. In our thesis we intend to generalize Boadlender's coloring game. In the respective chapters we introduce some relevant extensions and work out win- ning strategies for Alice for certain classes of graphs. In chapter 2 we extend the notion of game perfect graphs to the asymmetric coloring and introduce the asymmetric game perfect graphs. Further we investigate the two-person game considering circular colorings of weighted graphs in chapter 3. Chapter vii viii CONTENTS 4 is devoted to the circular game perfect graphs, an extension of the game per- fect graphs. Chapter 1 summarizes definitions and notations that will be used through- out the thesis. In chapter 2 we turn our attention to game perfect graphs. The first who dealt with this theory was S. D. Andres in [14]. He defined a graph G = (V; E) to be game perfect if and only if for every induced subgraph H ⊆ G it holds γ(H) = !(H), where !(H) := maxf n 2 N j Kn ⊆ Hg is the clique number of G. In our thesis we introduce the notion of asymmetric game perfect graphs. The asymmetric coloring differs from the ordinary coloring in the fact that the players are allowed to color several vertices in a row. Asymmetric coloring was introduced by H. A. Kierstead in [15] and received huge interest since it changes the ordinary game dramatically and it opens room for new research. As already mentioned we introduce asymmetric game perfect graphs. In par- ticular we consider the game for n 2 N and n ≥ 2, where Alice is allowed to color n vertices in a row and Bob one, the (n; 1)-coloring game. Contrary we also consider the game where Alice is allowed to color one vertex and Bob n vertices in a row, the (1; n)-coloring game. Precisely we determine the class of game perfect graphs for the (n; 1)-coloring game as well as for the (1; n)- coloring game for clique number 2. For this purpose we consider two varia- tions of the ordinary asymmetric game, namely the A-game and the B-game. In the A-game Alice starts the game and has the right to miss a turn. In the B-game Bob has these rights. We determine the class of asymmetric g-game perfect graphs with clique number 2 for g 2 fA; Bg for the (n; 1)-coloring game and the (1; n)-coloring game. In particular we show that only bipartite graphs can be g-game perfect and prove what conditions a bipartite graph must meet in order to be g-game perfect. Using these results we also determine the class of game perfect graphs with clique number 2 for the (n; m)-coloring game where n; m 2 N and n; m ≥ 2. In chapter 3 we introduce the circular two-person game on weighted graphs. Roughly speaking, this new game emphasizes a combination of Bodlaender's two-person game and the circular coloring of weighted graphs defined by W. CONTENTS ix A. Deuber and X. Zhu in [4] 1996, see definition 3.0.3. A graph Gw = (V; E; w) is called weighted if there exists a mapping w : V ! R+ which assigns to each vertex x 2 V a vertex-weight w(x) > 0. In contrast to Boadlender's game we consider graphs with vertex-weights. Furthermore, the vertices are not assigned to natural numbers but to arcs on a given circle C r in the Euclidean space R2 where r 2 R+ is the circumference. For (x; y) 2 E the arcs, denoted by fw(x) and fw(y), are not allowed to overlap in a feasible coloring. Further, for each vertex the length of the assigned arc equals its weight. Thus, the new game provides a generalization of the regular two-person game. We define w the circular game chromatic number of weighted graphs, denoted by γc(G ), as the least circumference r of Cr such that Alice has a winning strategy, where Gw is a weighted graph. First, some basic properties of the new parameter are investigated. In particular, we relate the circular game chromatic number of weighted graphs with the circular game chromatic number of ordinary graphs, w denoted by γc(G), and give conditions when γc(G) ≤ γc(G ) and γc(G) ≥ w γc(G ) hold. Afterwards, we turn our attention to the circular two-person game con- cerning relevant classes of weighted graphs. For all values of w we give up- per and lower bounds of the new parameter concerning the respective classes of graphs. Moreover, we consider some relevant distributions of the vertex- weights and work out winning strategies for Alice. First, we investigate w w γc(Kn ), where Kn is a weighted complete graph on n vertices, see definition 1.1.8. We work out a winning strategy for Alice, while restricting the weights by the mapping w : V ! fk; lg, for k; l 2 R+ and k =6 l. In particular, Alice's winning strategy has to be adapted to the frequencies that k and l appear in the complete graph. We proceed with the circular game chromatic number of weighted complete multipartite graphs, see definition 3.3.1. By applying tech- w niques of W. Lin and X. Zhu in [18], we determine γc(G ) for w : V ! fkg, k 2 R+.