The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture

Sheehan's conjecture (1977). Every 4-regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a different edge set).

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-lovasz-plummer-conjecture,
  title        = {The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/lovasz-plummer-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Sheehan's conjecture (1977).

Every $4$ -regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a different edge set). Sheehan's conjecture would settle the last open case of the question, raised by Smith's theorem for cubic graphs, of which regular Hamiltonian graphs have a second Hamiltonian cycle: Thomassen [Th98] proved it for all $r$ -regular graphs with $r \geq 300$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.LovaszPlummerConjecture.

theorem sheehan_conjecture :
    ∀ {V : Type} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj],
      (∀ v, G.degree v = 4) →
      ∀ (v : V) (c : G.Walk v v), IsHamiltonianCycle G c →
        ∃ (w : V) (c' : G.Walk w w), IsHamiltonianCycle G c' ∧
          c'.edges.toFinset ≠ c.edges.toFinset

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
[LP86] Lovász, L. and Plummer, M. D. (1986). Matching Theory. North-Holland.
[EKKKN11] Esperet, L., Kardoš, F., King, A. D., Král', D. and Norine, S. (2011). "Exponentially many perfect matchings in cubic graphs." Adv. Math. 227, pp. 1646--1664. arXiv:1012.2878
[Sh77] Sheehan, J. (1977). "The multiplicity of Hamiltonian circuits in a graph." In *Recent Advances in Graph Theory*, Academia, Prague, pp. 477--480.
[Th98] Thomassen, C. (1998). "Independent dominating sets and a second Hamiltonian cycle in regular graphs." J. Combin. Theory Ser. B 72, pp. 104--109.

Source and licence

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