The hat guessing number of random graphs with constant edge-chosen probability

TL;DR

This paper studies the hat guessing number in random graphs, showing it is typically close to the number of vertices for Erdős-Rényi graphs with constant edge probability, extending previous bounds.

Contribution

It extends existing bounds on the hat guessing number to all constant edge probabilities in Erdős-Rényi graphs, demonstrating it is nearly linear in the number of vertices.

Findings

HG(G) is at least n^{1 - o(1)} with high probability.

HG(G) is at most (1 - o(1))n with high probability.

Results hold for all constant p in (0,1).

Abstract

Let $G$ be a graph with $n$ vertices. The {\em hat guessing number} of $G$ is defined in terms of the following game: There are $n$ players and one opponent. The opponent will wear one of the $q$ hats of different colors on the player's head. At this time, the player can only see the player's hat color at the adjacent vertex, and communication between players is not allowed. Once players are assigned hats, each player must guess the color of his hat at the same time. If at least one player guesses right, then they will win collectively. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ different colors. Let $\mathcal{G}(n,p)$ denote the Erd\H{o}s-R\'{e}nyi random graphs with $n$ vertices and edge-chosen probability $ p\in(0,1)$. Alon-Chizewer and Bosek-Dudek-Farnik-Grytczuk-Mazur investigated the lower and upper bound for $HG(G)$ when $G\in \mathcal{G}(n,1/2)$, respectively. In this paper, we extends their results by showing that for any constant number $p$, we have $n^{1 - o(1)}\le HG(G) \le (1-o(1))n$ with high probability when $G\in \mathcal{G}(n,p)$.