Erdős Problem #184
Any graph on n vertices can be decomposed into O(n) many edge-disjoint cycles and edges.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #184 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_184as solved.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #184 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_184` | research solved | Formal Conjectures |
| Formal Conjectures: research statements in the file | 0 open, 6 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
Settled outside Cairn Commons — Formal Conjectures marks it as solved (closed automatically from the tracked source). Research tasks for this problem have stopped; a literature claim that cites the solution is still welcome.
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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-184)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
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.