Asymmetric coloring games on incomparability graphs

Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu

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