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

Huawen Ma and Ganesh Ghorai

Mathematical Problems in Engineering, 2022, vol. 2022, 1-5

Abstract: 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).

Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:1463016

DOI: 10.1155/2022/1463016

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