Digraph Coloring Games and Game-Perfectness Inaugural–Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨at zu K¨oln vorgelegt von Stephan Dominique Andres aus Geldern Verlag Dr. Hut, M¨unchen 2007 Berichterstatter: Prof. Dr. Ulrich Faigle Prof. Dr. Rainer Schrader Tag der m¨undlichen Pr¨ufung: 30.11.2007 Kurzzusammenfassung In dieser Arbeit wird die spielchromatische Zahl eines Digraphen als spiel- theoretische Variante der dichromatischen Zahl eingef¨uhrt. Dieser Begriff verallgemeinert die bekannte spielchromatische Zahl eines Graphen. Ein er- weitertes Modell ber¨ucksichtigt ebenfalls relaxierte F¨arbungen sowie asym- metrische Zugfolgen. Spielperfektheit wird als spieltheoretische Variante der Perfektheit eines Graphen definiert und auf Digraphen verallgemeinert. Obere und untere Schranken f¨ur die spielchromatische Zahl verschiedener Klassen von Digraphen werden untersucht. Im letzten Teil der Arbeit wer- den spielperfekte Digraphen mit kleiner Cliquenzahl charakterisiert, sowie all- gemeine Resultate ¨uber Spielperfektheit gezeigt. Einige Resultate wurden mit Hilfe eines Computerprogramms verifiziert, welches im Anhang diskutiert wird. Abstract In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game- perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix. Contents List of figures iii List of tables v About this thesis vii Acknowledgements ix 1 Introduction 1 1.1 Digraphs.............................. 2 1.2 Classesofgraphs ......................... 4 1.3 Surfaces .............................. 6 1.4 The dichromatic number of a digraph . 7 1.5 Adigraphcoloringgame . .. 8 1.6 Adigraphmarkinggame. 11 1.7 Examples ............................. 12 1.8 Previousresults .......................... 14 2 Forests 19 2.1 Coloringgamesondirectedforests. 19 2.2 Undirectedforests. .. .. 29 2.3 Forest-likestructures . 34 2.4 Orientationsofplanargraphs . 39 2.5 Openproblems .......................... 40 3 Lightness of digraphs in surfaces 43 3.1 Lightnessandweight . .. .. 43 3.2 The parameter M(S)....................... 46 3.3 Thestructureofdigraphsinsurfaces . 48 3.4 Thestructureofgraphsinsurfaces . 52 3.5 Applicationrange......................... 55 3.6 Tightnessofthebounds . 56 3.7 Application to coloring and marking games . 60 3.8 Anotherapplicationidea . 64 i ii CONTENTS 4 The incidence game chromatic number 67 4.1 k-degenerategraphs. 69 4.2 Somesimpleclassesofgraphs . 73 5 Game-perfect graphs 79 5.1 Perfectness and game-perfectness . 79 5.2 B-perfect graphs with clique number 2 . 82 5.3 A-perfect graphs with clique number 2 . 83 5.4 Bodlaender’soriginalversion. 87 5.5 Thegeneralcase ......................... 89 5.6 TowardsaStrongPerfectGraphTheorem . 91 5.7 B-perfectdigraphs ........................ 95 5.8 A-perfectdigraphs ........................ 95 5.8.1 Semiorientations of complete graphs . 96 5.8.2 Paths ........................... 97 5.8.3 Cycles ........................... 101 Appendix 105 A Complexity results 107 B Game-tree search 113 B.1 Theprogram ........................... 114 B.2 Runningtimeoftheprogram . 123 B.3 Computationalresults . 125 B.4 Acceleratingthealgorithm . 129 C Minor open questions 137 D Major open questions 139 Bibliography 141 Index 147 List of Figures 1.1 A graph, a simple digraph, and none of them . 3 1.2 Two ways of depicting a digraph . 3 1.3 AHalingraph........................... 5 1.4 The transitive tournament T4 .................. 13 2.1 Thepigeon-holeprinciple. 25 2.2 Possible configuration for w ................... 26 2.3 ThedigraphofLemma13 . .. 26 2.4 ThedigraphofProposition14 . 27 2.5 Vertex vik is added to the set of origin vertices . 30 2.6 Thefivecases ........................... 33 2.7 Adirected3-cycledforest . 35 2.8 Structure of the induced subtree of a T -component . 35 2.9 First case of splitting a T -component .............. 36 2.10 Second case of splitting a T -component. 36 2.11 Third case of splitting a T -component ............. 37 2.12 The main parts of the left half of D1 .............. 37 2.13 ThedigraphofTheorem23 . 41 3.1 A non-regular graph for which (3.1) is trivial . 44 3.2 The tree T(1,1)............................ 45 3.3 Subdividingthe‘fat’arcs. 49 3.4 Projective twisted dodekahedron . 58 3.5 Graphwithtorusidentification . 58 3.6 Digraph with Klein bottle identification . 58 3.7 Double-twisted double-clock . 59 3.8 Digraphwith63verticesand105arcs. 60 3.9 ThedigraphofTheorem43 . 63 4.1 Pairs of adjacent and non-adjacent incidences . 67 4.2 Arcs adjacent to (v,w) which may be activated . 70 4.3 The incidences adjacent to the incidence (v,vw)........ 71 4.4 A winning configuration for Bob on a cycle . 73 iii iv LIST OF FIGURES 4.5 StateofthegameafterBob’ssecondmove . 74 5.1 Two forbidden configurations for B-perfectgraphs . 81 5.2 Twographswithdiameter3 . 83 5.3 Two forbidden configurations for A-perfectgraphs . 85 5.4 The graph G2 ........................... 90 5.5 An interval graph which is not B-perfect ............ 90 5.6 An interval graph which is not A-perfect ............ 91 5.7 A broken wheel with clique number 3 . 92 5.8 Two more forbidden configurations for B-perfect graphs. 93 5.9 Two B-perfect graphs with clique number 4 . 94 5.10 g-perfect graphs the complements of which are not g-perfect . 95 5.11 Semiorientations of K3 ...................... 96 0 5.12 C~4 ................................. 97 5.13 The forbidden configuration F3,1 ................. 98 5.14 The forbidden configuration F3,2 ................. 98 5.15 The forbidden configuration F4 ................. 98 5.16 The forbidden configuration F5,1 ................. 98 5.17 The forbidden configuration F5,2 ................. 98 5.18 The forbidden configuration F7,1 ................. 98 5.19 The forbidden configuration F7,2 ................. 98 5.20 The forbidden configuration F8 ................. 98 5.21 The 47 A-perfectpaths...................... 99 5.22 8forbiddencycles. 100 5.23 The 14 A-perfect semiorientations of cycles . 101 A.1 Digraph for the formula f(x)................... 108 A.2 Simple digraph replacing an edge . 110 A.3 The configurations AT (n) and AL(n).............. 111 B.1 Thecube.............................. 127 B.2 Semiorientations of the Petersen graph . 128 List of Tables 3.1 Bounds for M (nonorientablecase) . 47 3.2 Bounds for M (orientablecase) . 48 3.3 Upper bounds for the lightness of certain graphs embeddable inanorientablesurface. 56 3.4 Upper bounds for the lightness of certain graphs embeddable inanonorientablesurface . 57 3.5 Upper bounds for the game coloring number of certain graphs embeddable in an orientable surface . 64 3.6 Upper bounds for the game coloring number of certain graphs embeddable in a nonorientable surface . 65 B.1 Running time versus density (2 colors, game gA)........ 125 B.2 Running time versus density (2 colors, game A) ........ 126 B.3 Running time versus density (3 colors, game gA)........ 126 B.4 Running time in a sparse case with the maximal number of colors127 B.5 Running time in the densest case with the maximal number of colors................................ 127 B.6 Computer results and CPU time for various instances . 129 B.7 Running time versus density (2 colors, game gA, improved al- gorithm).............................. 131 B.8 Running time versus density (2 colors, game A, improved algo- rithm) ............................... 131 B.9 Running time in a sparse case with the maximal number of colors(improvedalgorithm). 132 B.10 Running time of the improved algorithm on complete graphs . 133 B.11 Running time of the improved algorithm on semi-dense graphs 133 B.12 Results for the line graph of the Petersen graph (game gA with precoloring) ............................ 134 B.13 Results for the line graph of the Petersen graph (game gB with precoloring) ............................ 135 v vi LIST OF TABLES About this thesis In this thesis the game chromatic number of a digraph is introduced and ex- amined. This parameter is defined by the following non-cooperative 2-person game g. We are given a finite digraph D =(V, E), a finite color set C and two players, Alice and Bob. The players alternately color a vertex v of D with a color c of C which is selected in such a way that no in-neighbor of v is already colored with c. When no such move is possible any more, the game ends. If every vertex is colored at the end of the game, Alice wins, otherwise Bob. The game chromatic number of D is the smallest cardinality of a color set C for which Alice has a winning strategy for the game played with D and C. In this thesis graphs are always considered as digraphs with pairs of oppositely directed arcs. Then the game chromatic number of a digraph generalizes the well-known game chromatic number of a graph. After an introduction into the notions, a motivation, and a historical re- view in Chapter 1, we examine a generalization of the game on undirected and directed forests in Chapter 2. This generalization