For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge n/k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $k$. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize that result, proving that the conjecture is true for digraphs with independence number at most $(k+1)/2$.

中文翻译：

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Caccetta-Häggkvist 猜想的证明，用于具有小独立数的有向图

对于有向图 $G$ 和 $v \in V(G)$，令 $\delta^+(v)$ 是 $G$ 中 $v$ 的外邻居数。Caccetta-H\"{a}ggkvist 猜想指出，对于所有 $k \ge 1$，如果 $G$ 是 $n = |V(G)|$ 的有向图，使得 $\delta^+(v) \ge n/k$ 对于所有 $v \in V(G)$，则 G 包含一个长度最多为 $k$ 的有向环。在 [2] 中，N. Lichiardopol 证明了这个猜想对于具有独立性的有向图成立number 等于 2。在本文中，我们概括了该结果，证明对于独立数最多为 $(k+1)/2$ 的有向图，该猜想成立。