Vizing's conjecture (1968)

Vizing's conjecture (1968). For all finite simple graphs G and H, the domination number of the Cartesian (box) product satisfies γ(G square H) ≥ γ(G) γ(H).

Current state (tracked automatically)

Formal Conjectures marks the Lean statement vizing_conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `vizing_conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 3 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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-vizing-conjecture,
  title        = {Vizing's conjecture (1968)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/vizing-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Vizing's conjecture (1968).

For all finite simple graphs $G$ and $H$ , the domination number of the Cartesian (box) product satisfies $γ (G □ H) \geq γ (G) γ (H)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.VizingConjecture.

theorem vizing_conjecture :
    ∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
      (G : SimpleGraph α) (H : SimpleGraph β),
      G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Vi68] Vizing, V. G. (1968). "Some unsolved problems in graph theory." Uspekhi Mat. Nauk 23, pp. 117--134.
[ClSu00] Clark, W. E. and Suen, S. (2000). "An inequality related to Vizing's conjecture." Electron. J. Combin. 7, N4.
[SuTa12] Suen, S. and Tarr, J. (2012). "An improved inequality related to Vizing's conjecture." Electron. J. Combin. 19, P8.
[BDGHHKR12] Brešar, B., Dorbec, P., Goddard, W., Hartnell, B. L., Henning, M. A., Klavžar, S. and Rall, D. F. (2012). "Vizing's conjecture: a survey and recent results." J. Graph Theory 69, pp. 46--76.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.