Seymour's second-neighborhood conjecture

• Slug: seymour-second-neighborhood
• List: P37
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Directed graph theory
• Sources: Espuny Díaz et al. arXiv:2403.02842; 2026 small-order study arXiv:2601.21563
• Started: 2026-08-17

Statement

Every oriented graph is conjectured to contain a vertex with at least as many second out-neighbors as first out-neighbors. It remains widely open despite proofs for tournaments and several classes.

Tonight

A certified new small-order census, a new reducible configuration with an independently checkable certificate, or a proof for another hereditary class. Isolated random-graph statistics are an incomplete search. Fetch the 2026 small-order status before searching.

Tonight (2026-08-17): independently replayed Halkiewicz's labeled Pisa census through n=7, then classified every Pisa orientation of an 8-vertex graph up to undirected isomorphism. Exactly seven missing-degree types occur; three of them are irregular, which the n=7 pattern forbids. Certificates in compute/certs/. The conjecture itself is still open.