Result 157, Combinatorics

Graph coloring, clique minors, and Colin de Verdière invariants

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy χf(G)>h(G)\chi_f(G)\gt h(G), where h(G)h(G) is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound χf(G)≤μ(G)+1\chi_f(G)\le\mu(G)+1. In the positive direction, every finite nonempty graph satisfies χlist(G)≤Ch(G)\chi_{\mathrm{list}}(G)\le C h(G) for a universal constant C.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How tightly does a graph's coloring difficulty track the complete graphs hidden inside it? These manuscripts claim that two exact bounds fail even for fractional coloring, while a constant-factor bound survives for the stricter list-coloring problem.

What changes?

The manuscripts report arbitrarily large finite simple graphs with no three pairwise nonadjacent vertices whose fractional chromatic number exceeds their Hadwiger number. Fractional coloring minimizes the total weight on sets of nonadjacent vertices, covering each vertex with weight at least one. The Hadwiger number measures the largest complete graph obtainable through vertex deletions, edge deletions and edge contractions. Separate examples with the same independence restriction exceed the Colin de Verdière invariant, a numerical graph parameter, plus one, even fractionally.

What does that help mathematicians do?

Fractional coloring relaxes ordinary coloring, so these counterexamples would also invalidate the corresponding ordinary chromatic bounds. However, the third manuscript reports that every finite nonempty graph has list chromatic number at most C times its Hadwiger number, for an absolute integer C. List chromatic number is the minimum list size guaranteeing coloring from arbitrary vertex-specific color lists. Thus clique-minor size would retain universal constant-factor control even under these additional coloring constraints.

Are there practical applications?

The immediate value is foundational: the reported counterexamples would rule out exact coloring guarantees based on these graph parameters, while the positive result would preserve a weaker guarantee for vertex-specific color choices. The abstracts provide no coloring algorithm or numerical value for C, so practical performance does not follow from the stated bound.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

A counterexample to Hadwiger's conjecture

September 23, 2026 104 pages

We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-Hadwigers-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Hadwiger's conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Hadwigers-conjecture-September-23-2026}},
  year = {2026}
}

A counterexample to the Colin de Verdière chromatic conjecture

September 23, 2026 131 pages

We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to the Colin de Verdi{\`e}re chromatic conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026}},
  year = {2026}
}

A linear list-coloring bound in terms of the Hadwiger number

September 23, 2026 42 pages

We prove that every finite nonempty graph G satisfies χlist(G)≤Ch(G)\chi_{\mathrm{list}}(G)\le C h(G) for an absolute integer C, where h(G)h(G) is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026,
  author = {{OpenAI}},
  title = {{A linear list-coloring bound in terms of the Hadwiger number}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf}{OAI:A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/157.md.

Graph coloring, clique minors, and Colin de Verdière invariants

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Linear List Hadwiger conjecture asks for a universal linear bound on list chromatic number in terms of clique-minor size. The formalization proves that there is one integer C≥1C\ge1 such that every finite nonempty simple graph GG satisfies χlist(G)≤Ch(G)\chi_{\mathrm{list}}(G)\le C h(G), where h(G)h(G) is the largest order of a clique minor. The constant is independent of the graph.

Comparator links

Result Comparator statement
Linear list-coloring bound in the Hadwiger number ListHadwiger.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.