Seymour’s Second Neighborhood Conjecture for m-Free Oriented Graphs

Abstract

Let $\left(D=\left(V,E\right)\right)$ be an oriented graph with minimum out-degree ${\delta }^{+}$ . For $x\in V\left(D\right)$ , let $d_D^{+}\left(x\right)$ and $d_D^{++}\left(x\right)$ be the out-degree and second out-degree of $x$ in $D$ , respectively. For a directed graph $D$ , we say that a vertex $x\in V\left(D\right)$ is a Seymour vertex if $\left(d_D^{++}\left(x\right)\ge d_D^{+}\left(x\right)\right)$ . Seymour in 1990 conjectured that each oriented graph has a Seymour vertex. A directed graph $D$ is called $m$ -free if there are no directed cycles with length at most $m$ in $D$ . A directed graph $D=\left(V,E\right)$ is called $k$ -transitive if, for any directed $xy$ -path of length $k$ , there exists $\left(\left(x,y\right)\in E\right)$ . In this paper, we show that (1) each $\left({\delta }^{+}-2\right)$ -free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in $m$ -free and $\left(2m+2\right)$ -transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).