Seymour-tight orientations

TL;DR

This paper studies Seymour-tight orientations, a special class where the first and second neighborhoods of each vertex are equal, exploring their properties, constructions, and implications for Seymour's and Sullivan's conjectures.

Contribution

It introduces the concept of Seymour-tight orientations, proves their closure under lexicographic products, and connects them to algebraic structures and conjecture counterexamples.

Findings

Seymour-tight orientations are closed under lexicographic products.

Counterexamples to Seymour's conjecture can be constructed using these orientations.

Certain Seymour-tight orientations correspond to Cayley digraphs of abelian groups.

Abstract

We investigate `almost counterexamples' to Seymour's second neighbourhood conjecture. In what we call Seymour-tight orientations, the size of the first neighbourhood of each vertex equals the size of its second neighbourhood. We give several examples and constructions. Specifically, we prove that the class of Seymour-tight orientations is closed under taking (generalized) lexicographic products. Moreover, the lexicographic product of a putative counterexample to Seymour's second neighbourhood conjecture and a Seymour-tight orientation is again a counterexample. Using lexicographic products, we show that if the conjecture is false, then there exist counterexamples that are close to regular tournaments, and moreover that any digraph occurs as an induced subgraph of a counterexample. We then use this same machinery to construct special putative counterexamples to Sullivan's conjecture. The inherent symmetry of these orientations give access to an algebraic perspective. Seymour-tight orientations that are also Cayley digraphs correspond to special pairs of critical sets in groups, which connects potentially to additive combinatorics. We use Kemperman's theorem to characterize those Seymour-tight orientations that are the Cayley digraph of an abelian group.