Graph colorings with restricted bicolored subgraphs: II. The graph coloring game

TL;DR

This paper investigates the graph coloring game, establishing bounds on the game chromatic number based on properties of bicolored subgraphs and applying these results to graph products.

Contribution

It introduces bounds on the game chromatic number for graphs with bounded game coloring numbers and extends these bounds to Cartesian and strong graph products.

Findings

Bounded game coloring number implies bounded game chromatic number.

Cartesian product of graphs with bounded game coloring number also has bounded game chromatic number.

Provides an upper bound for the game chromatic number of the strong product of two graphs.

Abstract

We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, answering a question of X. Zhu. We also obtain an upper bound on the game chromatic number of the strong product $G_1 \boxtimes G_2$ of two graphs.