@MISC{Krzywkowski11hatproblem, author = {Marcin Krzywkowski}, title = {HAT PROBLEM ON ODD CYCLES }, year = {2011} }

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Abstract

The topic is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of a win. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The hat problem on a graph was solved for trees and for the cycle on four vertices. Then Uriel Feige conjectured that for any graph the maximum chance of success in the hat problem is equal to the maximum chance of success for the hat problem on the maximum clique in the graph. He provided several results that support this conjecture, and solved the hat problem for bipartite graphs and planar graphs containing a triangle. We make a step towards proving the conjecture of Feige. We solve the hat problem on all cycles of odd length. Of course, the maximum chance of success for the hat problem on the cycle on three vertices is three fourths. We prove that the hat number of every odd cycle of length at least five is one half, which is consistent with the conjecture of Feige.