Guessing games on triangle-free graphs
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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 Markström introduced a method to bound the value of the guessing number from below using the fractional clique cover number kappa_f(G). In particular they showed gn(G) >= |V(G)| - kappa_f(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.
Cameron , P J , Dang , A & Riis , S 2016 , ' Guessing games on triangle-free graphs ' , Electronic Journal of Combinatorics , vol. 23 , no. 1 , P1.48 . < https://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p48 >
Electronic Journal of Combinatorics
© 2016, the Author(s). This work is made available online in accordance with the publisher’s policies. This is the final published version of the work, which was originally published at https://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p48
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