Seymour's second neighbourhood conjecture: random graphs and reductions

A longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed \(p\in[0,1/2)\), a.a.s. every orientation of \(G(n,p)\) satisfies Seymour's conje...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Alberto Espuny Díaz, Girão, António, Granet, Bertille, Kronenberg, Gal
Format: Article
Language:English
Subjects:
Reduction
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed \(p\in[0,1/2)\), a.a.s. every orientation of \(G(n,p)\) satisfies Seymour's conjecture (as well as a related conjecture of Sullivan). This improves on a recent result of Botler, Moura and Naia. Moreover, we show that \(p=1/2\) is a natural barrier for this problem, in the following sense: for any fixed \(p\in(1/2,1)\), Seymour's conjecture is actually equivalent to saying that, with probability bounded away from \(0\), every orientation of \(G(n,p)\) satisfies Seymour's conjecture. This provides a first reduction of the problem. For a second reduction, we consider minimum degrees and show that, if Seymour's conjecture is false, then there must exist arbitrarily large strongly-connected counterexamples with bounded minimum outdegree. Contrasting this, we show that vertex-minimal counterexamples must have large minimum outdegree.
ISSN:2331-8422