Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hat{\mu}$, arising from the hat guessing game. The parameter $\hat{\mu}$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hat{\mu}$ of cliques, paths, and cycles.

翻译：考虑以下猜帽子游戏。一只熊坐在图 $G$ 的每个顶点上，一个恶魔给每只熊戴上由 $h$ 种颜色中一种染色的帽子。每只熊只能看到其邻居的帽子颜色。仅基于此信息，每只熊需要猜测 $g$ 种颜色，如果其帽子颜色包含在猜测中，则算作猜对。如果对于任何帽子排列，至少有一只熊猜对，则熊方获胜。我们引入一个新参数——分数帽子色数 $\hat{\mu}$，它源于此猜帽子游戏。参数 $\hat{\mu}$ 与之前研究过的帽子色数相关。我们揭示了猜帽子游戏与图的独立多项式之间的惊人联系。这一联系使我们能够在多项式时间内计算弦图的分数帽子色数，将分数帽子色数界定为 $G$ 的最大度函数，并计算出团、路径和环的 $\hat{\mu}$ 精确值。