Hat Guessing Numbers of Strongly Degenerate Graphs

Abstract

Assume players are placed on vertices of a graph . The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of available colors. The players can see the hat of each of their neighbors in but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer such that there is a winning strategy for the players is denoted by , and this is called the hat guessing number of . Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from to 40 and further derive upper bounds on the hat guessing number for any class of -free graphs with bounded expansion, such as the class of -free planar graphs; more generally, for -free graphs with bounded Hadwiger number or without a -subdivision; and for Erdős–Rényi random graphs with constant average degree.