戴帽子的熊与独立多项式
Bears with Hats and Independence Polynomials
AI总结：
本文研究图上的猜帽游戏，引入分数帽子色数$\hatμ$，并揭示其与图独立多项式的联系，从而实现多项式时间内计算弦图参数、以最大度界定参数及求出团、路径和环的精确值。
摘要：
考虑以下猜帽游戏。每只熊坐在图$G$的每个顶点上，恶魔给每只熊戴上一顶由$h$种颜色之一染色的帽子。每只熊只能看到其邻居的帽子颜色。仅基于此信息，每只熊必须猜测$g$种颜色，若其帽子颜色包含在其猜测中，则该熊猜对。如果对于任何帽子排列，至少有一只熊猜对，则熊获胜。我们引入了一个新参数——分数帽子色数$\hatμ$，它源于猜帽游戏。该参数$\hatμ$与之前研究过的帽子色数相关。我们提出了猜帽游戏与图的独立多项式之间令人惊讶的联系。这种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，用$G$的最大度函数来界定分数帽子色数，并计算团、路径和环的$\hatμ$精确值。
英文摘要：
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 $\hatμ$, arising from the hat guessing game. The parameter $\hatμ$ 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 $\hatμ$ of cliques, paths, and cycles.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
