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The Colin de Verdiere Chromatic Conjecture

Theorem 1.1: there are graphs of arbitrarily large order with α(G)≤2\alpha(G)\le2 and μ(G)+1<∣V(G)∣/2≤χ(G)\mu(G)+1<|V(G)|/2\le\chi(G); the fractional chromatic number also exceeds μ(G)+1\mu(G)+1. The proof shows every well-signed matrix on the graph with one negative eigenvalue has rank above ∣V(G)∣/2+1|V(G)|/2+1, without using the Strong Arnold Property, so the Lovasz-Schrijver parameter κ\kappa also satisfies κ(G)+1<∣V(G)∣/2\kappa(G)+1<|V(G)|/2. Since h≤μ+1h\le\mu+1, these graphs also violate Hadwiger's inequality, by a route independent of the connected-matching companion. Nothing is claimed for small μ\mu.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Colin de Verdiere Chromatic Conjecture · Colin de Verdiere chromatic conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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VibeMathed record: The Colin de Verdiere Chromatic Conjecture evidence for this event

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Theorem 1.1: there are graphs of arbitrarily large order with α(G)≤2\alpha(G)\le2 and μ(G)+1<∣V(G)∣/2≤χ(G)\mu(G)+1<|V(G)|/2\le\chi(G); the fractional chromatic number also exceeds μ(G)+1\mu(G)+1. The proof shows every well-signed matrix on the graph with one negative eigenvalue has rank above ∣V(G)∣/2+1|V(G)|/2+1, without using the Strong Arnold Property, so the Lovasz-Schrijver parameter κ\kappa also satisfies κ(G)+1<∣V(G)∣/2\kappa(G)+1<|V(G)|/2. Since h≤μ+1h\le\mu+1, these graphs also violate Hadwiger's inequality, by a route independent of the connected-matching companion. Nothing is claimed for small μ\mu. parent of this event

The Colin de Verdiere Chromatic Conjecture parent of this event

A counterexample to the Colin de Verdiere chromatic conjecture evidence for this event

The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture evidence for this event

Colin de Verdiere 1990, JCTB 50 evidence for this event

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The Colin de Verdiere Chromatic Conjecture — Mathematical Frontier Network