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Seymour’s second-neighborhood conjecture

The conjecture was known for important special graph classes.

Progress records 1

Understand the problem ↓

1 record · How we rate progress

Understanding the problem

Read the background

The conjecture was known for important special graph classes. The unrestricted oriented-graph target removes those structural restrictions.

Loops, parallel arcs, and opposite arc pairs are excluded. Second neighbors are at distance exactly two, so vertices already reached in one step are not counted again.

Why it matters

If correct, resolves the second-neighborhood target for the full stated oriented-graph class.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • Loops, parallel arcs, and opposite arc pairs are excluded. Second neighbors are at distance exactly two, so vertices already reached in one step are not counted again.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

Website timeline

Publication dates and editorial changes.