Game coloring the Cartesian product of graphs

Abstract

Abstract This article proves the following result: Let G and G ′ be graphs of orders n and n ′, respectively. Let G * be obtained from G by adding to each vertex a set of n ′ degree 1 neighbors. If G * has game coloring number m and G ′ has acyclic chromatic number k , then the Cartesian product G □ G ′ has game chromatic number at most k ( k + m − 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008