An improved bound on Seymour's second neighborhood conjecture

TL;DR

This paper improves the known bound related to Seymour's second neighborhood conjecture, establishing that there exists a vertex with a second out-neighborhood at least 71.55% the size of its first, marking a significant progress after over twenty years.

Contribution

The paper presents the first enhancement of the constant factor in Seymour's second neighborhood conjecture in over twenty years.

Findings

Existence of a vertex with second out-neighborhood at least 71.55% of the first.

First improvement to the bound in over two decades.

Advances understanding of neighborhood sizes in oriented digraphs.

Abstract

Seymour's celebrated second neighborhood conjecture, now more than thirty years old, states that in every oriented digraph, there is a vertex $u$ such that the size of its second out-neighborhood $N^{++}(u)$ is at least as large as that of its first out-neighborhood $N^+(u)$. In this paper, we prove the existence of $u$ for which $|N^{++}(u)| \ge 0.715538 |N^+(u)|$. This result provides the first improvement to the best known constant factor in over two decades.