发表机构
Université Paris Cité（巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明每个二色数足够大的竞赛图包含具有大有向围长和大二色数的子有向图，并探讨了用底层图围长替代的情形，该情形蕴含Erdős-Hajnal猜想且对特定竞赛图族成立。
AI 中文摘要
在20世纪60年代，Erdős和Hajnal猜想：每个色数足够大的图都包含一个具有大围长（最小环的长度）和大色数的子图。在本文中，我们证明了每个二色数足够大的竞赛图都包含一个具有大有向围长（最小有向环的长度）和大二色数的子有向图。我们研究了将围长替换为（底层图的）围长时的相同命题。我们表明该命题蕴含Erdős和Hajnal的猜想，并针对一类特定的竞赛图证明了该命题。
英文摘要
In the 1960s, Erdős and Hajnal conjectured that every graph with sufficiently large chromatic number contains a subgraph of large girth (size of a smallest cycle) and large chromatic number. In this paper, we prove that every tournament with sufficiently large dichromatic number contains a subdigraph of large digirth (size of a smallest directed cycle) and large dichromatic number. We investigate the same statement when replacing digirth by girth (of the underlying graph). We show that it implies the conjecture of Erdős and Hajnal, and prove it for a particular family of tournaments.
