Indicated coloring game on Cartesian products of graphs

Abstract

The indicated coloring game is played on a simple graph G by two players, with a fixed set C of colors. In each round of the game Ann indicates an uncolored vertex, and Ben colors it using a color from C, obeying just the proper coloring rule. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent this. The minimum cardinality of the set of colors C for which Ann has a winning strategy is called the indicated chromatic number, χi(G), of a graph G. In this paper, we prove that the indicated chromatic number of the Cartesian product G□Kn,m is equal to 3 if χi(G)=3. We also prove that χi(G□T)=χ(G), where G is a block graph and T is a tree. Indicated colorings in some other classes of Cartesian products of graphs are also studied. The investigations lead us to propose a Sabidussi-type equality as an open problem.