Impartial coloring games

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function

232^3 Quantified Boolean Formula Games and Their Complexities

Consider QBF, the Quantified Boolean Formula problem, as a combinatorial game ruleset. The problem is rephrased as determining the winner of the game where two opposing players take turns assigning values to boolean variables. In this paper, three common variations of games are applied to create seven new games: whether each player is restricted to where they may play, which values they may set variables to, or the condition they are shooting for at the end of the game. The complexity for determining which player can win is analyzed for all games. Of the seven, two are trivially in P and the other five are PSPACE-complete. These varying properties are common for combinatorial games; reductions from these five hard games can simplify the process for showing the PSPACE-hardness of other games.Comment: 14 pages, 0 figures, for Integers 2013 Conference proceeding

Games for One, Games for Two: Computationally Complex Fun for Polynomial-Hierarchical Families

In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in are thought of as 0-turn games, as 1-turn “puzzle” games, and in general ₖ as -turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of -turn CIRCUIT SATISFIABILITY games, whose ₖ-completeness is assumed from prior work (we briefly justify this assumption on intuitive grounds, but no proof is given). In the second half, we introduce and explore the properties of a novel family of games called the -turn GRAPH 3-COLORABILITY games. By embedding boolean circuits in proper graph 3-colorings, we construct reductions from -turn CIRCUIT SATISFIABILITY games to -turn 3-COLORABILITY games, thereby showing that -turn 3-COLORABILITY is ₖ-complete Finally, we conclude by discussing possible future generalizations of this work, vis-à-vis extending arbitrary -complete puzzles to interesting ₖ-complete games

The Orthogonal Colouring Game

International audienceWe introduce the Orthogonal Colouring Game, in which two players alternately colour vertices (from a choice of m ∈ N colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the colouring. Each player aims to maximise her score, which is the number of coloured vertices in the copy of the graph she owns. The main result of this paper is that the second player has a strategy to force a draw in this game for any m ∈ N for graphs that admit a strictly matched involution. An involution σ of a graph G is strictly matched if its fixed point set induces a clique and any non-fixed point v ∈ V (G) is connected with its image σ(v) by an edge. We give a structural characterisation of graphs admitting a strictly matched involution and bounds for the number of such graphs. Examples of such graphs are the graphs associated with Latin squares and sudoku squares

HABILITATION A DIRIGER DES RECHERCHES Graphes et jeux combinatoires

On considère généralement que la théorie des graphes est née au 18e siècle, et qu'elle connaît un essor significatif depuis les années 1960. L'avènement de la théorie des jeux combinatoires est quant à lui plus récent (fin des années 1970). Ce domaine reste alors moins exploré dans la littérature, et de nombreuses études sur des techniques générales de résolution sont toujours actuellement en cours de construction. Dans ce mémoire, je propose plusieurs tours d'horizons à propos de problématiques bien ciblées de ces deux domaines.Dans un premier temps, je m'interroge sur la complexité des règles de jeux de suppression de tas. Il s'avère que dans la littérature, la complexité d'un jeu est souvent définie comme la complexité algorithmique d'une stratégie gagnante. Cependant, il peut aussi avoir du sens de s'interroger sur la nature des règles de jeu. Un premier pas dans cette direction a été fait avec l'introduction du concept de jeu dit invariant. On notera au passage que certains résultats obtenus ont mis en exergue des liens entre combinatoire des mots et stratégie gagnante d'un jeu. Dans un deuxième chapitre, j'aborde les jeux sous l'angle des graphes. Deux aspects sont considérés:* Un graphe peut être vu comme un support de jeu. Le cas du jeu de Nim et ses variantes sur les graphes y est examiné.* Certaines problématiques standard de théorie des graphes peuvent être transformées dans une version ludique. C'est d'ailleurs un objet d'étude de plus en plus prisé par la communauté. Nous détaillerons le cas des jeux de coloration sommet.Enfin, le dernier chapitre se concentre sur deux nouvelles variantes de problématiques issues de la théorie des graphes: le placement de graphes et les colorations distinguantes. J'en profite pour faire un état de l'art des principaux résultats sur ces deux domaines

Impartial coloring games

International audienceColoring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k > 0 colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function