Guessing Games on Triangle-Free Graphs

The guessing game introduced by Riis [Electron. J. Combin. 2007] 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 Markström [Electron. J. Combin. 2011] introduced a method to bound the value of the guessing number from below using the fractional clique cover number κf (G). In particular they showed gn(G) > |V (G)| − κf (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.