A Remark on the Second Neighborhood Problem

TL;DR

This paper investigates Seymour's second neighborhood conjecture, proving that certain classes of digraphs, including tournaments and digraphs missing a matching, contain vertices with the second neighborhood property, advancing understanding of the conjecture.

Contribution

The paper introduces the concept of 'good' digraphs and proves that every feed vertex in a tournament has the SNP, also extending results to digraphs missing a matching.

Findings

Every feed vertex of a tournament has the SNP.

Existence of vertices with SNP in digraphs missing a matching.

Refinement of previous proofs using good digraphs.

Abstract

Seymour's second neighborhood conjecture states that every simple digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). We define "good" digraphs and prove a statement that implies that every feed vertex of a tournament has the SNP. In the case of digraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due to Fidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices with SNP.