关于Caccetta-Häggkvist猜想的Aharoni彩虹推广
On Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture
摘要：
对于有向图$G$和$v \in V(G)$，令$δ^+(v)$为$G$中$v$的外邻居数。Caccetta-Häggkvist猜想指出，对于所有的$k \ge 1$，如果$G$是一个有向图，其$n = |V(G)|$使得$δ^+(v) \ge k$对于所有的$v \in V(G)$，则G包含一个长度至多为$\lceil n/k \rceil$的有向圈。在[2]中，Aharoni提出了一个关于这个猜想的推广，即一个在$n$顶点上的简单边着色图，其$n$色类的大小为$k$，其彩虹圈的长度至多为$\lceil n/k \rceil$ .在本文中，我们证明了这个猜想，如果每个颜色类的大小$Ω(k \log k)$。
英文摘要：
For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $\lceil n/k \rceil$. In [2], Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. In this paper, we prove this conjecture if each color class has size $Ω(k \log k)$.
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。
