Abstract
Let D = V , E be an oriented graph with minimum out-degree δ + . For x ∈ V D , let d D + x and d D + + 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 ∈ V D is a Seymour vertex if d D + + x ≥ d D + 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 x y -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 2 m + 2 -transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).
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