Seymour's Second Neighborhood Conjecture for Subsets of Vertices

TL;DR

This paper explores a subset-based approach to Seymour's Second Neighborhood Conjecture, proving an equivalent formulation and improving bounds on neighborhood ratios in certain directed graphs.

Contribution

It introduces a subset perspective that provides new insights and improves bounds related to the second neighborhood conjecture in directed graphs.

Findings

Equivalent formulation of Seymour's conjecture for subsets of vertices

Counterexamples have bounded size related to minimum degree

Improved lower bounds for neighborhood ratios in m-free graphs

Abstract

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. In this note, we put forward a conjecture that we prove is actually equivalent: every oriented simple graph contains a subset of vertices $S$ whose second neighborhood is at least as large as its first. This subset perspective gives some insight into the original conjecture. For example, if there is a counterexample to the second neighborhood conjecture with minimum degree $\delta$, then there exists a counterexample on at most ${\delta + 1 \choose 2}$ vertices. Given a vertex $v$, let $d_1^+(v)$ and $d_2^+(v)$ be the size of its first and second neighborhoods respectively. A digraph is $m$-free if there is no directed cycle on $m$ or fewer vertices. Let $\lambda_m$ be the largest value such that every $m$-free graph contains a vertex $v$ with $d_2^+(v) \geq \lambda_m d_1^+(v)$. The second neighborhood conjecture implies $\lambda_m = 1$ for all $m \geq 2$. Liang and Xu provided lower bounds for all $\lambda_m$, and showed that $\lambda_m \to 1$ as $m \to \infty$. We improve on Liang and Xu's bound for $m \geq 3$ using this subset perspective.