The Arc-Weighted Version of the Second Neighborhood Conjecture

2012/12/09 by Tyler Seacrest, Seacrest, Tyler
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Graph Labeling and Dimension Problems #math.CO

paper · pdf · doi:10.48550/arxiv.1212.1883

10 pages, 3 figures

arxiv created 2012/12/09 · openalex publication_date 2012/12/09 · arxiv updated 2012/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher. One extension of the conjecture that has been used by several researchers is to consider vertex-weighted digraphs. In this paper we introduce a version of the conjecture for arc-weighted digraphs. We prove the conjecture in the special case of arc-weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.

Related

• Seymour's Second Neighborhood Conjecture for Subsets of Vertices
• A generalisation of Seymour's second neighbourhood conjecture
• Seymour's second-neighborhood conjecture from a different perspective
• About the second neighborhood conjecture for tournaments missing two stars or disjoint paths
• Seymour's second neighbourhood conjecture for quasi-transitive oriented graphs