The Number of Seymour Vertices in Random Tournaments and Digraphs

TL;DR

This paper investigates the prevalence of Seymour vertices in random tournaments and digraphs, demonstrating that such vertices are almost surely abundant in these random structures, supporting aspects of Seymour's distance two conjecture.

Contribution

It provides probabilistic evidence that random tournaments and digraphs almost surely contain many Seymour vertices, advancing understanding of Seymour's conjecture in random graph models.

Findings

Almost surely many Seymour vertices in random tournaments.

Even more Seymour vertices in general random digraphs.

Supports probabilistic aspects of Seymour's distance two conjecture.

Abstract

Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements along lines suggested by Seymour's conjecture, proving that almost surely there are a "large" number of Seymour vertices in random tournaments and "even more" in general random digraphs.