On the Hat Guessing Number of Graphs

The hat guessing number $HG(G)$ of a graph $G$ on $n$ vertices is defined in terms of the following game: $n$ players are placed on the $n$ vertices of $G$, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players 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 hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. In this note we construct a planar graph $G$ satisfying $HG(G)=12$, settling a problem raised in \cite{BDFGM}. We also improve the known lower bound of $(2-o(1))\log_2 n$ for the typical hat guessing number of the random graph $G=G(n,1/2)$, showing that it is at least $n^{1-o(1)}$ with probability tending to $1$ as $n$ tends to infinity. Finally, we consider the linear hat guessing number of complete multipartite graphs.