New constructions and bounds for Winkler's hat game

Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given colour; they do not see their own colour, but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colours to their hats, at least one player guesses their own hat colour correctly. In this paper, we discover some new classes of graphs which allow a winning strategy, thus answering some of the open questions in Butler et al. We also derive upper bounds on the maximal number of possible hat colours that allow for a winning strategy for a given graph.