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 ∈ V, the existence of a directed path of length k from u to v implies that (u, v) ∉ 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.