Hat guessing number and guaranteed subgraphs

TL;DR

The paper investigates when the hat guessing number $ ext{HG}(G)$ is bounded by graph structure. It proves that forbidding a fixed tree subgraph yields a constant upper bound, and that graphs with cycles bounded in length also have a constant bound, providing explicit formulas. The methods combine a modified multi-guess hat guessing variant, block decompositions, and treedepth analysis, leveraging Sylvester-type recurrences to bound the two-guess version on bounded treedepth graphs. These results deepen understanding of how local forbidden configurations constrain global guessing dynamics and suggest avenues for further exploration of general forbidden subgraphs and girth-related questions.

Abstract

The hat guessing number of a graph is a parameter related to the hat guessing game for graphs introduced by Winkler. In this paper, we show that graphs of sufficiently large hat guessing number must contain arbitrary trees and arbitrarily long cycles as subgraphs. More precisely, for each tree $T$, there exists a value $N = N(T)$ such that every graph that does not contain $T$ as a subgraph has hat guessing number at most $N$, and for each integer $c$, there exists a value $N' = N'(c)$ such that every graph with no cycle of length greater than $c$ has hat guessing number at most $N'$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof