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Level A · Machine-checkable Hard Graph theory P-erdos-184

Erdős Problem #184

Any graph on n vertices can be decomposed into O(n) many edge-disjoint cycles and edges.

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-184,
  title        = {Erdős Problem #184},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-184}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Any graph on vertices can be decomposed into many edge-disjoint cycles and edges.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«184».

theorem erdos_184 :
    ∃ f : ℕ → ℝ,
      (f =O[atTop] fun n : ℕ ↦ (n : ℝ)) ∧
      ∀ {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V),
      ∃ (D : Finset G.Subgraph),
        (∀ H ∈ D, IsCycleOrEdge H.coe) ∧
        IsDecomposition G D ∧
        (D.card : ℝ) ≤ f (Fintype.card V)

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/184. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_184.variants.covering — In [Er71] Erdős suggests that only n-1 many cycles and edges are required if we do not require them to be edge-disjoint.

References

  • erdosproblems.com/184
  • [BM22] Bucić, M. and Montgomery, R., Towards the Erdős-Gallai Cycle Decomposition Conjecture. arXiv:2211.07689 (2022).
  • [CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures Algorithms (2014), 608-626.
  • [EGP66] Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112.
  • [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

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.