Abstract
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 $\mathrm{gn}(G)$ 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 $\kappa_f(G)$. In particular they showed $\mathrm{gn}(G) \geq |V(G)| - \kappa_f(G)$. 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 $50$.
Highlights
The motivation of developing guessing games [17] comes from the study of a specific class of problems in network coding [3], namely multiple unicast network coding
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
It is worth noting that guessing games were the main ingredients in Riis’ proof of the invalidity of two conjectures raised by Valiant [19] in circuit complexity in which he asked about the optimal Boolean circuit for a Boolean function. (See [17].)
Summary
The motivation of developing guessing games [17] comes from the study of a specific class of problems in network coding [3], namely multiple unicast network coding. A multiple unicast network is a communication network in which each sender has a unique receiver that wishes to obtain messages from it Such a network can be represented by a directed acyclic graph where senders, receivers and routers are vertices in the graph and channels are directed edges between vertices. We show that there are triangle-free undirected graphs where the guessing numbers of these graphs can not be computed using the fractional clique cover method developed by Christofides and Markstrom in [8]. This gives counterexamples to their conjecture about the optimal guessing strategy based on fractional clique cover in [8].
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