Erdős Problem #628 — open problem in graph theory

Erdős Problem #628

Let G be a graph with chromatic number k containing no K_k. If a,b≥ 2 and a+b=k+1 then must there exist two disjoint subgraphs of G with chromatic numbers ≥ a and ≥ b respectively?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-628,
  title        = {Erdős Problem #628},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-628}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $G$ be a graph with chromatic number $k$ containing no $K_{k}$ . If $a, b \geq 2$ and $a + b = k + 1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\geq a$ and $\geq b$ respectively?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«628».

theorem erdos_628 (V : Type*) [Fintype V] (G : SimpleGraph V) (k : ℕ)
    (hG_chrom : G.chromaticNumber = (k : ℕ∞))
    (hG_clique : G.CliqueFree k)
    (a b : ℕ) (ha : a ≥ 2) (hb : b ≥ 2) (hab : a + b = k + 1) :
    ∃ (s : Set V),
      (G.induce s).chromaticNumber ≥ (a : ℕ∞) ∧
      (G.induce sᶜ).chromaticNumber ≥ (b : ℕ∞)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/628. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/628
[BKPS09] Balogh, József and Kostochka, Alexandr V. and Prince, Noah and Stiebitz, Michael, The Erdős-Lovász Tihany conjecture for quasi-line graphs. Discrete Math. (2009), 3985-3991.
[BrJu69] Brown, W. G. and Jung, H. A., On odd circuits in chromatic graphs. Acta Math. Acad. Sci. Hungar. (1969), 129-134.
[Er68b] Erdős, P., Problem 2. Theory of Graphs (1968), 361.
[So22] Song, Zi-Xia, A survey on the Erdős-Lovász Tihany conjecture. Adv. Math. (China) (2022), 259--274.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.