arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1812.09752math.COcs.ITmath.IT
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图的帽子猜测数

The hat guessing number of graphs

Noga Alon, Omri Ben-Eliezer, Chong Shangguan, Itzhak Tamo

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25 pages, minor revision, to appear in JCTB
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摘要：

考虑下面的猜帽游戏：$n$玩家被放置在图的$n$顶点上，每个玩家戴着一顶帽子，帽子的颜色是从一组$q$可能的颜色中任意选择的。每个玩家可以看到他的邻居的帽子颜色，但不能看到他自己的帽子颜色。所有的玩家都被要求根据预先确定的猜测策略和他们看到的帽子颜色同时猜测他们自己的帽子颜色，其中不允许他们之间进行交流。给定一个图$G$，它的帽子猜测数${\rm{HG}}(G)$是最大整数$q$，使得存在一个猜测策略，保证对$q$可能颜色的任何帽子分配至少有一个正确的猜测。在2008年，Butler等人提出了一个问题：完全二部图$K_{n,n}$的帽子猜测数是否至少是$n$的某个固定的正（分数）幂。我们肯定地回答了这个问题，证明了对于足够大的$n$，完全$r$ -部图$K_{n,\ldots,n}$满足${\rm{HG}}(K_{n,\ldots,n})=Ω(n^{\frac{r-1}{r}-o(1)})$。我们的猜测策略是基于概率结构和其他组合的想法，并可以扩展到显示${\rm{HG}}(\vec{C}_{n,\ldots,n})=Ω(n^{\frac{1}{r}-o(1)})$，其中$\vec{C}_{n,\ldots,n}$是一个有向$r$ -循环的爆破，并在有向图中，每个玩家只看到他的外邻居的帽子颜色。

英文摘要：

Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, 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. Given a graph $G$, its hat guessing number ${\rm{HG}}(G)$ 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 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, the complete $r$-partite graph $K_{n,\ldots,n}$ satisfies ${\rm{HG}}(K_{n,\ldots,n})=Ω(n^{\frac{r-1}{r}-o(1)})$. Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that ${\rm{HG}}(\vec{C}_{n,\ldots,n})=Ω(n^{\frac{1}{r}-o(1)})$, where $\vec{C}_{n,\ldots,n}$ is the blow-up of a directed $r$-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.

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。

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