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OpenAI reports: a counterexample to Hadwiger’s conjecture

The manuscript reports arbitrarily large finite simple graphs with independence number at most two and chromatic number greater than their largest clique-minor order.

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Understanding the problem

Read the background

The manuscript reports arbitrarily large finite simple graphs with independence number at most two and chromatic number greater than their largest clique-minor order. It also claims violation of the ordinary fractional-coloring weakening.

Why it matters

If correct, disproves a central graph-minor/coloring conjecture and its stated fractional weakening.

What counts as progress

A sourced result with the exact hypotheses, baseline, and AI role stated; verification is assessed separately.

What to watch for

  • Compare the precise theorem and hypotheses with the previous baseline.
  • Read the checking scope and retain human and prior-work attribution.

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

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