The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs

Oriented coloring: complexity and approximation

International audienceThis paper is devoted to an oriented coloring problem motivated by a task assignment model. A recent result established the NP-completeness of deciding whether a digraph is k-oriented colorable; we extend this result to the classes of bipartite digraphs and circuit-free digraphs. Finally, we investigate the approximation of this problem: both positive and negative results are devised

The graph distance game and some graph operations

In the graph distance game, two players alternate in constructing a max- imal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this paper we examine the distance game for various graph operations: the join, the corona and the lexicographic product of graphs. We provide general bounds and exact results for special graphsPostprint (published version

Refined activation strategy for the marking game

AbstractThis paper introduces a new strategy for playing the marking game on graphs. Using this strategy, we prove that if G is a planar graph, then the game colouring number of G, and hence the game chromatic number of G, is at most 17

The game Grundy number of graphs

Given a graph G = (V;E), two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbours. Alice's goal is to minimize the total number of colours used in the game, and Bob's goal is to maximize it. The game Grundy number of G is the number of colours used in the game when both players use optimal strategies. It is proved in this paper that the maximum game Grundy number of forests is 3, and the game Grundy number of any partial 2-tree is at most 7

Digraph Coloring Games and Game-Perfectness

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

Spieltheoretische Kantenfärbungsprobleme auf Wäldern und verwandte Strukturen

This diploma thesis discusses graph colouring games, as introduced by Bodlaender [3], in a more general setting. The main results are: The directed and undirected game chromatic indices of the class of forests of maximum degree D are D+1, for D=3, D=5, and D>5. The directed game chromatic indices of the class of forests of maximum degree 2 are 2 or 3, depending on whether passing is allowed for Alice in the underlying game. The method of decomposition of independent subtrees is extended from edge colouring games to node colouring games and leads to new proofs resp. new results concerning the undirected resp. directed (new) game chromatic numbers of the class of forests. These numbers are 4 (3) resp. 3 (3). The results mentioned so far are also true for infinite instead of finite graphs. Some general properties of graph colouring games are examined. A list of important open questions can be found at the end

On the Oriented Game Chromatic Number \Lambda

The game chromatic number of G, denoted by gcn(G), is then defined as the least cardinality of a color set X for which Alice has a winning strategy. This parameter is well defined since we clearly have O/(G) ^ gcn(G) ^ jV (G)j