Guessing Games on Triangle-free Graphs

TL;DR

This paper investigates the guessing game on triangle-free graphs, revealing that some such graphs, including the Higman-Sims graph, have guessing numbers exceeding the fractional clique cover bound, challenging previous assumptions.

Contribution

It demonstrates that the fractional clique cover bound does not always accurately predict the guessing number for triangle-free graphs, providing new insights into graph guessing game bounds.

Findings

Higman-Sims graph has guessing number between 77 and 78.

Fractional clique cover bound is not tight for some triangle-free graphs.

Counterexamples challenge previous bounds on guessing numbers.

Abstract

The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstrom introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.