A bound for the game chromatic number of graphs

Abstract

We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k( k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs. In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the game chromatic number of a planar graph is at most 30, which improves the previous known upper bound for the game chromatic number of planar graphs.