On Seymour's Second Neighborhood Conjecture of m-free Digraphs

TL;DR

This paper investigates Seymour's Second Neighborhood Conjecture for m-free digraphs, providing an approximate relation between second out-neighborhoods and out-degrees that approaches equality as m increases.

Contribution

It introduces an approximate bound for m-free digraphs, extending and refining previous results related to Seymour's conjecture.

Findings

Existence of a vertex v with second out-neighborhood proportional to out-degree

The proportionality constant λ_m approaches 1 as m increases

Generalization and improvement of known results in digraph theory

Abstract

This paper gives an approximate result related to Seymour's Second Neighborhood conjecture, that is, for any $m$-free digraph $G$, there exists a vertex $v\in V(G)$ and a real number $\lambda_m$ such that $d^{++}(v)\geq \lambda_m d^+(v)$, and $\lambda_m \rightarrow 1$ while $m \rightarrow +\infty$. This result generalizes and improves some known results in a sense.