  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 $(k+1)/2$.
