On the degree of trees with Game Chromatic Number 4

Abstract

The coloring game is played by Alice and Bob on a finite graph $G$. They take turns properly coloring the vertices with $t$ colors. The goal of Alice is to color the input graph with t colors, and Bob does his best to prevent it. If at any point there exists an uncolored vertex without available color, then Bob wins; otherwise Alice wins. The game chromatic number $\chi_g(G)$ of $G$ is the smallest number $t$ such that Alice has a winning strategy. In 1991, Bodlaender showed the smallest tree $T$ with $\chi_g(T)$ equal to $4$, and in 1993 Faigle et al. proved that every tree $T$ satisfies the upper bound $\chi_g(T) \leq 4$. The stars $T = K_{1,p}$ with $p\geq 1$ are the only trees satisfying $\chi_{g}(T)=2$; and the paths $T= P_n$, $n\geq 4$, satisfy $\chi_{g}(T)=3$. Despite the vast literature in this area, there does not exist a characterization of trees with $\chi_g(T) = 3$ or $4$. We answer a question about the required degree to ensure $\chi_g(T) = 4$, by exhibiting infinitely many trees with maximum degree 3 and game chromatic number 4.