A note on the second neighborhood problem for k-anti-transitive digraphs with large girth

Seymour’s 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 . The girth of a digraph is the length of its shortest directed cycle. In this paper, we prove that if D is a k -anti-transitive digraph whose girth is larger than k − 4 , 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.