Hat guessing with proper colorings

TL;DR

This paper introduces a new variant of the hat guessing game on graphs where the adversary provides a proper coloring, and determines the maximum number of colors for which a vertex-dependent guessing strategy guarantees at least one correct guess.

Contribution

It establishes the hat guessing number for complete graphs and trees under this proper coloring constraint, and provides bounds and exact values for small graphs, along with conjectures.

Findings

Hat guessing number of complete graphs on n vertices is 2n - 1.

Hat guessing number of all trees with n ≥ 3 vertices is 4.

Exact values and bounds for small graphs, with improved estimates for book graphs.

Abstract

We initiate the study of the hat guessing number of a graph where the adversary is only allowed to provide a proper coloring of the graph. This is the largest number $q$ for which there is a guessing strategy on each vertex that only depends on its neighborhood, such that for every proper coloring of the graph with $q$ colors at least one vertex guesses its color correctly. In this variation, we prove that the hat guessing number of the complete graphs on $n$ vertices is $2n - 1$, which is roughly twice the classical hat guessing number of the complete graph. Our winning strategy is related to finding perfect matchings between the middle layers of the boolean poset of dimension $2n - 1$. We prove that the hat guessing number of all trees on $n \geq 3$ vertices is equal to $4$. We derive some general upper and lower bounds for all graphs and give improved estimates for book graphs. Using our results and an ILP formulation of the problem, we determine the exact hat guessing number for all graphs on at most $4$ vertices, give bounds on graphs on $5$ vertices, and propose general conjectures.