AI 中文总结
针对未解决的Bermond-Thomassen猜想，证明了其松弛版本，即最小出度至少为$2k-1$的有向图含$k$个顶点不相交的环，每个环可通过反转一条弧变为有向，该界紧并解答了2021年研讨会问题。
AI 中文摘要
著名的Bermond-Thomassen猜想指出，每个最小出度至少为$2k-1$的有向图都包含$k$个顶点不相交的有向环。尽管该猜想于1981年提出，但对于所有$k\neq4$的情况仍未解决。我们证明了该猜想的一个松弛版本：每个最小出度至少为$2k-1$的有向图$D$都包含$k$个顶点不相交的环，每个环要么是有向的，要么通过反转其一条弧即可变为有向的。该界是紧的，且回答了Cames van Batenburg在2021年“熵压缩及相关方法”在线研讨会上提出的问题。
英文摘要
The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint cycles, each of which either is directed or can be made directed by reversing one of its arcs. This bound is sharp and answers a question raised by Cames van Batenburg during the online workshop "Entropy Compression and Related Methods" in $2021$.
Comments10 pages, 1 figure
