Result 106, Theoretical computer science

Hardness of coloring three-colorable graphs

It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0<δ<1/30\lt \delta\lt 1/3, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size δn\delta n otherwise, where n is the number of vertices.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

Coloring a graph means assigning colors to its vertices so that connected vertices differ. An unreviewed manuscript claims that even when three colors suffice, finding a coloring with any fixed larger palette is NP-hard.

What changes?

The manuscript reports a deterministic polynomial-time reduction from 3SAT, a Boolean satisfiability problem. For every fixed real number delta strictly between zero and one-third, it produces simple, unweighted graphs with n vertices. Satisfiable formulas produce three-colorable graphs; unsatisfiable formulas produce graphs whose independent sets all have fewer than delta times n vertices. An independent set is a group of vertices with no edges between them.

What does that help mathematicians do?

In any coloring with c colors, some color class contains at least one-c-th of all vertices and is an independent set. Choosing delta below one divided by c therefore excludes such a coloring in the unsatisfiable case. Consequently, for every fixed integer c at least three, a polynomial-time algorithm guaranteed to color three-colorable graphs with c colors would imply P equals NP.

Are there practical applications?

The immediate value is foundational: separating the existence of a small coloring from the ability to find one efficiently, even with extra colors available. The claimed barrier concerns worst-case guarantees, not whether heuristics work on particular graphs. It also leaves open what algorithms can achieve when the allowed number of colors grows with graph size.

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.

Manuscript

Hardness of finding large independent sets in three-colorable graphs

September 24, 2026 21 pages Main result formalized in Lean

We prove that, for every fixed 0<δ<1/30\lt \delta\lt 1/3, it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026,
  author = {{OpenAI}},
  title = {{Hardness of finding large independent sets in three-colorable graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026.pdf}{OAI:Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/106.md.

Hardness of coloring three-colorable graphs

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

Scope

The formalized result shows hardness of finding large independent sets even under a three-colorability promise. For every fixed 0<δ<1/30<\delta<1/3, a deterministic polynomial-time reduction maps binary 3SAT formulas to nonempty finite simple graphs. Satisfiable formulas produce three-colorable graphs; unsatisfiable formulas produce graphs whose largest independent set has fewer than δ\delta times the number of vertices. The complete adjacency-matrix output is included in the runtime bound.

Comparator links

Result Comparator statement
Independent-set hardness in three-colorable graphs IndependentSets.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.