Hat Guessing Numbers of Degenerate Graphs

TL;DR

This paper investigates the hat guessing number of degenerate graphs, establishing exponential lower bounds and introducing a new method for upper bounds, leaving the boundedness question open.

Contribution

It proves exponential lower bounds for the hat guessing number in degenerate graphs and presents a new technique for deriving upper bounds.

Findings

Existence of d-degenerate graphs with hat guessing number at least 2^{2^{d-1}}

Introduction of a new method for upper bounding the hat guessing number

Open problem on whether the hat guessing number is bounded by degeneracy

Abstract

Recently, Farnik asked whether the hat guessing number $\text{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\text{HG}(G)\ge 2^d$ is possible. We show that for all $d\ge 1$ there exists a $d$-degenerate graph $G$ for which $\text{HG}(G) \ge 2^{2^{d-1}}$. We also give a new general method for obtaining upper bounds on $\text{HG}(G)$. The question of whether $\text{HG}(G)$ is bounded as a function of $d$ remains open.