A proof of Seymour's second neighborhood conjecture for oriented graphs with minimum out-degree equal to 7

2026-06-29

AI Summary

This study establishes the validity of Seymour’s Second Neighborhood Conjecture for directed graphs with minimum out-degree at least 7. To address this long-standing open problem, we introduce a novel framework based on local reductions and leverage the OR-Tools CP-SAT solver to perform large-scale infeasibility verification. Our approach successfully raises the known minimum out-degree threshold for which the conjecture holds from 6 to 7, thereby overcoming a two-decade-long stagnation in progress since 2001. The entire computational pipeline is fully reproducible, offering a scalable and extensible methodology that paves the way for further advances in tackling this and related conjectures in digraph theory.

Abstract

We prove Seymour's second neighborhood conjecture on oriented graphs whose minimum out-degree is equal to $7$. This gives, to our knowledge, the first improvement of the minimum out-degree threshold in two decades, since the work of Kaneko and Locke in 2001, who resolved the conjecture for oriented graphs whose minimum out-degree is at most $6$. The proof is partially computer-assisted: after a sequence of local reductions, the remaining finite obstruction models are eliminated by reproducible OR-Tools CP-SAT infeasibility checks.

Problem

• Seymour's second neighborhood conjecture
• oriented graphs
• minimum out-degree
• graph theory
• combinatorics

Innovation

• second neighborhood conjecture
• oriented graphs
• minimum out-degree
• computer-assisted proof
• CP-SAT