On the hat guessing number of a planar graph class

TL;DR

This paper establishes upper bounds on the hat guessing number for outerplanar and layered planar graphs using novel decomposition and hypergraph density techniques.

Contribution

Introduces a new vertex decomposition method and applies hypergraph density theorems to bound the hat guessing number of planar graph classes.

Findings

Outerplanar graphs have a hat guessing number less than 2^125000.

Layered planar graphs also have bounded hat guessing number.

New techniques connect hypergraph density with graph guessing parameters.

Abstract

The hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erd\H{o}s for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than $2^{125000}$. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered planar graph has bounded hat guessing number.