An Upper Bound on the Hat Guessing Number of Graphs

The hat guessing number $HG(G)$ of a graph is defined by the following game: each player is placed on a vertex and assigned a hat with one of $k$ colors. Each vertex can see only the hat color of the other vertices it is connected to in $G$. All vertices guess, simultaneously, the color of their own hat. The hat guessing number $HG(G)$ is the largest $k$ such that the players can guarantee that at least one of them guesses correctly. In this paper, we show a general bound on the hat guessing number of a graph $G$ as a function of its order $n$ and its maximum degree $\Delta$. This is the first nontrivial upper bound on $HG(G)$ as a function of $\Delta$ and $n$ when $\Delta \geq \frac{n}{e}$. From this result we also obtain that the hat guessing number of the random graph $G_{n,1/2}$ is at most asymptotically $cn$ for $c\sim 0.809$, and that graphs with maximum degrees of $(1-\varepsilon )n$ for fixed $\varepsilon>0$ cannot have $HG(G)=(1-o(1))n$.