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

Let D=V,E be an oriented graph with minimum out-degree δ+. For x∈VD, let dD+x and dD++x be the out-degree and second out-degree of x in D, respectively. For a directed graph D, we say that a vertex x∈VD is a Seymour vertex if dD++x≥dD+x. 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=V,E is called k-transitive if, for any directed xy-path of length k, there exists x,y∈E. In this paper, we show that (1) each δ+−2-free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in m-free and 2m+2-transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).