Hat guessing number and guaranteed subgraphs

TL;DR

This paper investigates the relationship between the hat guessing number of a graph and its subgraph structure, showing that large guessing numbers imply the presence of complex subgraphs like trees and long cycles.

Contribution

It establishes bounds on the hat guessing number based on the absence of specific subgraphs, linking graph structure to the guessing game parameter.

Findings

Graphs with large hat guessing number contain all trees as subgraphs.

Graphs with no long cycles have bounded hat guessing number.

The paper provides explicit bounds related to subgraph exclusion.

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 $N'$.