Game List Colouring of Graphs

Abstract

We consider the two-player game defined as follows. Let $(G,L)$ be a graph $G$ with a list assignment $L$ on its vertices. The two players, Alice and Bob, play alternately on $G$, Alice having the first move. Alice's goal is to provide an $L$-colouring of $G$ and Bob's goal is to prevent her from doing so. A move consists in choosing an uncoloured vertex $v$ and assigning it a colour from the set $L(v)$. Adjacent vertices of the same colour must not occur. This game will be called game list colouring. The game choice number of $G$, denoted by ch$_g(G)$, is defined as the least $k$ such that Alice has a winning strategy for any $k$-list assignment of $G$. We characterize the class of graphs with ch$_g(G)\le 2$ and determine the game choice number for some classes of graphs.