The Berge–Fulkerson conjecture

The Berge–Fulkerson conjecture (Fulkerson 1971). Every bridgeless cubic graph has six perfect matchings such that every edge lies in exactly two of them.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement berge_fulkerson_conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `berge_fulkerson_conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	3 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-berge-fulkerson-conjecture,
  title        = {The Berge–Fulkerson conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/berge-fulkerson-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

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

The Berge–Fulkerson conjecture (Fulkerson 1971).

Every bridgeless cubic graph has six perfect matchings such that every edge lies in exactly two of them.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.BergeFulkersonConjecture.

theorem berge_fulkerson_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
      IsCubic G → G.IsBridgeless →
      ∃ M : Fin 6 → G.Subgraph, IsPerfectMatchingCover G 6 2 M

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.

Variants

berge_fulkerson_conjecture.variants.berge — Berge's conjecture: five perfect matchings covering every edge. Every bridgeless cubic graph has five perfect matchings whose union is the whole edge set.
berge_fulkerson_conjecture.variants.fan_raspaud — The Fan–Raspaud conjecture (1994). Every bridgeless cubic graph has three perfect matchings with no edge common to all three.

References

Wikipedia
[Fu71] Fulkerson, D. R. (1971). "Blocking and anti-blocking pairs of polyhedra." Math. Programming 1, pp. 168--194.
[Se79] Seymour, P. D. (1979). "On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte." Proc. London Math. Soc. 38, pp. 423--460.
[Ma11] Mazzuoccolo, G. (2011). "The equivalence of two conjectures of Berge and Fulkerson." J. Graph Theory 68, pp. 125--128.
[FR94] Fan, G. and Raspaud, A. (1994). "Fulkerson's conjecture and circuit covers." J. Combin. Theory Ser. B 61, pp. 133--138.
[Pe91] Petersen, J. (1891). "Die Theorie der regulären graphs." Acta Math. 15, pp. 193--220.

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.