Seymour's Second Neighborhood Conjecture

Prove that for every oriented directed graph G = (V, E) with no loops and no pairs of opposite arcs, there exists a vertex v in V such that the size of its second out-neighborhood N++(v) is at least the size of its first out-neighborhood N+(v), i.e., |N++(v)| ≥ |N+(v)|. Here, N+(v) denotes the set of out-neighbors of v, and N++(v) denotes the set of vertices reachable from v by a directed path of length two.

Background

The conjecture, posed by Paul Seymour and first published by Dean and Latka (1995), asks for the existence of a vertex in every oriented graph whose second neighborhood is at least as large as its first. It has been a central problem in digraph theory, with special cases proven (e.g., tournaments) and numerous partial results, but the general case has historically been regarded as difficult.

This paper presents a claimed proof via the GLOVER framework, yet the text explicitly identifies the conjecture as such and situates it within the historical context of being an open problem prior to the paper's approach. The formal statement in Conjecture 1.1 serves as the canonical formulation of the problem to be resolved.

References

Conjecture 1.1. (Seymour's Second Neighborhood Conjecture). For every oriented graph G = (V, E), there exists a verter v E V such that |N++(v)| ≥ |N+(v)|, where Nt and N++ represent the first and second neighborhoods of v respectively.

— An Algorithmic Approach to Finding Degree-Doubling Nodes in Oriented Graphs  (2501.00614 - Glover, 2024) in Conjecture 1.1, Section 1 (Introduction)

The following is known as Seymour's second neighbourhood conjecture. Every orientation $G$ contains at least one vertex $v$ such that $|\NGi{2}{G}{v}| \geq |\NGi{1}{G}{v}|$.

— Seymour-tight orientations  (2603.29626 - Guo et al., 31 Mar 2026) in Conjecture (Seymour), Section 1 (Introduction)