Hat guessing number for the class of planar graphs is at least 22

Abstract

We analyze the following version of the deterministic Hats game. We have a graph G, and a sage resides at each vertex of G. When the game starts, an adversary puts on the head of each sage a hat of a color arbitrarily chosen from a set of k possible colors. Each sage can see the hat colors of his neighbors but not his own hat color. All of the sages are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The strategy is winning if it guarantees at least one correct individual guess for every color assignment.In a modified version of the hat guessing game each sage makes s⩾1 guesses. Given a graph G and integer s⩾1, the hat guessing number HGs(G) is the maximal number k such that there exists a winning strategy.In this paper, we present new constructors, i.e. theorems that allow built winning strategies for the sages on different graphs. Using this technique, we calculate the hat guessing number HGs(G) for paths and “petunias”, and present a planar graph G for which HG1(G)≥22.