Hat Guessing on Books and Windmills

TL;DR

This paper determines the exact hat-guessing numbers for specific graph classes, including book graphs, windmill graphs, and the bipartite graph K_{3,3}, advancing understanding of this graph invariant.

Contribution

It provides exact values for the hat-guessing number for certain graphs, improving bounds and resolving open cases in the study of this graph invariant.

Findings

Exact hat-guessing number for book graphs with many pages

Hat-guessing number of K_{3,3} is 3, less than the upper bound

Determined the hat-guessing number for most windmill graphs

Abstract

The hat-guessing number is a graph invariant defined by Butler, Hajiaghayi, Kleinberg, and Leighton. We determine the hat-guessing number exactly for book graphs with sufficiently many pages, improving previously known lower bounds of He and Li and exactly matching an upper bound of Gadouleau. We prove that the hat-guessing number of $K_{3,3}$ is $3$, making this the first complete bipartite graph $K_{n,n}$ for which the hat-guessing number is known to be smaller than the upper bound of $n+1$ of Gadouleau and Georgiou. Finally, we determine the hat-guessing number of windmill graphs for most choices of parameters.