Tuza's conjecture — open problem in graph theory

Tuza's conjecture

Tuza's conjecture (1981): every finite graph G satisfies τ(G) ≤ 2ν(G), where τ(G) is its triangle covering number and ν(G) is its triangle packing number. The bound would be sharp, for example for K_4 and K_5.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement tuza_conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `tuza_conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 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-tuza-conjecture,
  title        = {Tuza's conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/tuza-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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Disputed

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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

Tuza's conjecture (1981): every finite graph $G$ satisfies $τ (G) \leq 2 ν (G)$ , where $τ (G)$ is its triangle covering number and $ν (G)$ is its triangle packing number. The bound would be sharp, for example for $K_{4}$ and $K_{5}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.TuzaConjecture.

theorem tuza_conjecture {V : Type} [Fintype V] (G : SimpleGraph V) :
    triangleCoverNumber G ≤ 2 * trianglePackingNumber G

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
[Tu90] Tuza, Zs. (1990). "A conjecture on triangles of graphs." Graphs Combin. 6, pp. 373--380.
[Ha99] Haxell, P. E. (1999). "Packing and covering triangles in graphs." Discrete Math. 195, pp. 251--254.

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.