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。图$G$的每个顶点上坐着一只熊，恶魔给每只熊戴上一顶由$h$种颜色之一染色的帽子。每只熊仅能看到其邻居的帽子颜色。基于这一信息，每只熊必须猜测$g$种颜色，若其帽子颜色包含在其猜测中则猜对。若对于任意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$\hat{\mu}$，该参数源于猜帽子游戏。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，将分数帽子色数以$G$的最大度函数形式进行界定时，并精确计算团、路径和环的$\hat{\mu}$值。