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Math-2603 · mathematics · Combinatorics

Colin de Verdière chromatic-number conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does every finite simple graph G satisfy $chi(G)<=\mu (G)+1$, where mu is the Colin de Verdière invariant?

Short English statement

Does every finite simple graph G satisfy $chi(G)<=\mu (G)+1$, where mu is the Colin de Verdière invariant?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2603-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2603

Colin de Verdière chromatic obstruction: The primary theorem has alpha(G)<=2 and mu(G)+1<|V(G)|/2<=chi(G) for arbitrarily large finite graphs. These finite simple graph witnesses contradict chi<=mu+1.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Colin de Verdière chromatic-number conjecture

OpenAI's claim

Colin de Verdière chromatic obstruction: The manuscript claims finite graphs whose fractional chromatic number is greater than $\mu(G)+1$, and therefore whose ordinary chromatic number violates the same bound. Such examples exist at arbitrarily large orders.

Mathematical statement of the claimed result

Mathematical claim 1.
The claimed counterexamples are finite simple graphs of arbitrarily large order $m$ with \[\alpha(G)\le2,\qquad\mu(G)+1<m/2\le\chi(G),\] where $\mu(G)$ is the Colin de Verdière spectral graph invariant. The fractional-coloring corollary strengthens the obstruction to \[\chi_f(G)\ge m/\alpha(G)\ge m/2>\kappa(G)+1\ge\mu(G)+1\ge h(G).\] Here $\kappa$ is its strong-Arnold-property-free spectral relaxation, and $h$ is the largest complete-minor order. In particular both $\chi\le\mu+1$ and $\chi_f\le h$ are refuted. A further corollary gives $\chi_{\mathrm{list}}(G)\le C(\mu(G)+1)$ with an absolute $C$.

Outline of the argument

Colin de Verdière chromatic obstruction: Allowed channel images are thinned while preserving the primal distribution, supplying enough compatible pairs for the conflict estimates. A Boolean moment argument restricts low-rank matrix representations. Entropy and finite sampling then produce graphs for which every well-signed matrix with a single negative eigenvalue has too much rank to support the proposed chromatic inequality.

Formal verification

Family 157 describes Lean coverage for the list-coloring upper bound with a universal multiple of clique-minor size; neither chromatic counterexample is part of this selection.

Colin de Verdière chromatic obstruction: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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