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arXiv · 2606.30588

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

Arpan Sadhukhan·R. B. Sandeep·Sagnik Sen
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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.

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Arpan Sadhukhan, R. B. Sandeep, Sagnik Sen. 2026-06-29. A proof of Seymour's second neighborhood conjecture for oriented graphs with minimum out-degree equal to 7. https://arxiv.org/abs/2606.30588

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