A note on the second neighborhood problem for $k$-anti-transitive and $m$-free digraphs

Seymour Second Neighborhood Conjecture (SSNC) asserts that every finite oriented graph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is called a Seymour vertex. A digraph $D = (V, E)$ is $k$-anti-transitive if for every pair of vertices $u, v \in V$, the existence of a directed path of length $k$ from $u$ to $v$ implies that $(u, v) \notin E$. An $m$-free digraph is digraph having no directed cycles with length at most $m$. In this paper, we prove that if $D$ is $k$-anti-transitive and $(k-4)$-free digraph, then $D$ has a Seymour vertex. As a consequence, a special case of Caccetta-Haggkvist Conjecture holds on 7-anti-transitive oriented graphs. This work extends recently known results.