The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc $uv$ in a tournament $T$ is the set of vertices that form a directed triangle with arc $uv$. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erd\H{o}s and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.

翻译：有向图的色数是指覆盖其顶点集所需的最小导出无圈子有向图数量；相应地，竞赛图的色数是指覆盖其顶点集所需的最小传递子竞赛图数量。竞赛图 $T$ 中弧 $uv$ 的邻域定义为与该弧构成有向三角形的顶点集合。我们证明：若竞赛图中每条弧的邻域均具有有界色数，则整个竞赛图亦具有有界色数。该结论可更广泛地适用于具有有界独立数的定向图，且我们将证明从竞赛图推广至这类稠密有向图类。作为应用，我们证明了 El-Zahar 与 Erdős 的一个猜想与 Nguyen、Scott 及 Seymour 近期关于高色数图与竞赛图结构关联的猜想之间的等价性。