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arXiv 1702.01607math.CO
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着色比赛：从本地到全球

Coloring tournaments: from local to global

Ararat Harutyunyan, Tien-Nam Le, Stéphan Thomassé, Hehui Wu

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7 pages, no figure
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摘要：

有向图$D$的\emph{chromatic number}是对$D$的顶点着色所需的最小颜色数，使得$D$的每个颜色类都导出一个无环子有向图。因此，竞赛图$T$的色数是覆盖$T$顶点集的传递子竞赛图的最小数目。在本文中，我们表明，比赛显着简单，比图着色。事实上，虽然无向图可以完全“局部简单”（每个邻域是一个独立集），并有大的色数，我们表明，局部简单的比赛确实是简单的。特别地，存在函数$f$，使得如果竞赛图$T$中的每个顶点的外邻域具有至多$c$的色数，则$T$具有至多$f(c)$的色数。这回答了Berger等人的一个问题。

英文摘要：

The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the minimum number of transitive subtournaments which cover the vertex set of $T$. We show in this paper that tournaments are significantly simpler than graphs with respect to coloring. Indeed, while undirected graphs can be altogether "locally simple" (every neighborhood is a stable set) and have large chromatic number, we show that locally simple tournaments are indeed simple. In particular, there is a function $f$ such that if the out-neighborhood of every vertex in a tournament $T$ has chromatic number at most $c$, then $T$ has chromatic number at most $f(c)$. This answers a question of Berger et al.

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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