Guessing Games on Triangle-Free Graphs

Abstract

The guessing game introduced by Riis[17] 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 [8] introduced a method to bound the value of the guessing number from below using the fractional clique cover number κ f (G). In particular they showed gn(G) ≥ |V (G)| − κ 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.Remark 5.10. For a more extensive list of computations of ranks of matrices A + kI over F q for q = 2, 3, 5, 7 see EBasis.zip at https://www.eecs.qmul.ac.uk/~smriis/.Problem 5.11. Find the exact value of the guessing number of the strongly regular graphs considered here.

Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability

“…The guessing number of loopless digraphs was thoroughly investigated in [5], [36], [6], [32], [37], [38];…”

“…Until now, however, the only known linearly solvable undirected graphs are vertex-full. Based on the results in [38], we can construct the first example of a linearly solvable undirected graph which is not vertex-full. Firstly, for two digraphs G 1 and G 2 on disjoint vertex sets of sizes n 1 and n 2 respectively, their bidirectional union is G := G 1∪ G 2 where G 1 and G 2 are linked by all possible edges between them.…”

“…There exists an undirected graph which is linearly solvable, and yet it is not vertex-full.Proof: Let G 1 := E 6 denote the empty graph on n 1 := 6 vertices and with linear guessing number 0 for any q. Let G 2 := C denote the Clebsch graph: C has n 2 := 16 vertices, independence number α(C) = 5, and g L (C, 3) ≥ 10[38]. Since C is triangle-free but not vertex-full, it is not linearly solvable as we shall see below.…”

High-rate storage codes on triangle-free graphs

“…Essentially the same problem, in a different guise, appears in a line of works devoted to guessing games on graphs, e.g., [10,8], which has developed independently of both the storage codes and index coding problems. That these groups of problems are largely equivalent was realized in a number of papers, and the historical development is detailed in [2] from the index codes' perspective.…”

On the possible values of the entropy of undirected graphs

“…Some graphs, notably the odd cycles and their complements, have noninteger entropies ; moreover, exhibits a small graph (10 vertices) whose entropy cannot be expressed as a rational number with a small denominator. It is also worth noting that some famous simple triangle‐free graphs, such as the Clebsch graph of the Higman‐Sims graph, have an entropy above .…”