Seymour's Second Neighborhood in 3-Free Digraphs

Abstract

In this paper, we consider Seymour's Second Neighborhood Conjecture in 3-free digraphs, and prove that for any 3-free digraph $D$, there exists a vertex say $v$, such that $d$⁺⁺($v$) ≥ $⌊λd^+(v)⌋$, $λ$ = 0.6958 · · · . This slightly improves the known results in 3-free digraphs with large minimum out-degree.