A counterexample to the Colin de Verdière chromatic conjecture

OpenAI

The manuscript claims arbitrarily large finite graphs G with independence number at most two and μ(G)+1<|V(G)|/2≤χ(G), contradicting the proposed inequality χ(G)≤μ(G)+1.

Sources for this record

What was already known

The inequality was known in low-μ cases. The new family uses large graphs and a rank obstruction for well-signed matrices.

AI involvement AI-led

OpenAI credits an internal model with the manuscript and proof. No named human proof checker or explanation is documented.

The cited release credits an internal OpenAI model with producing the result and the manuscript credits OpenAI as author; the record does not establish model autonomy from task selection or human direction.

Importance Notable

It refutes a recognized spectral chromatic conjecture.

Limitations

The construction does not give a small-μ counterexample. No completed check of the graph construction or universal matrix-rank argument is documented; the companion Hadwiger construction is separate.

Technical verification Reported

The manuscript supplies the construction and rank argument, but no completed check of either by a named human or machine is documented. The release’s formalization catalogue lists no selected top-line theorem.

Human understanding Few

No named person explains the central argument in the reviewed sources. Few is the evidence default, not a finding that people cannot understand it.

Correction and publication history