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Level A · Machine-checkable Hard Graph theory P-erdos-583

Erdős Problem #583

Every connected graph on n vertices can be partitioned into at most ⌈ n/2⌉ edge-disjoint paths. A problem of Erdős and Gallai.

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-583,
  title        = {Erdős Problem #583},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-583}},
  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

Every connected graph on vertices can be partitioned into at most edge-disjoint paths.

A problem of Erdős and Gallai.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«583».

theorem erdos_583 {V : Type*} [Fintype V] (G : SimpleGraph V) (hG : G.Connected) :
    ∃ D : Finset G.Subgraph,
      (∀ H ∈ D, IsPathSubgraph H) ∧
      IsDecomposition G D ∧
      D.card ≤ ⌈(Fintype.card V : ℚ) / 2⌉₊

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/583. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/583
  • [AnBa23] Anto, Nevil and Basavaraju, Manu, Gallai's path decomposition for 2-degenerate graphs. Discrete Math. Theor. Comput. Sci. (2023), Paper No. 16, 11.
  • [BBB21] A. Blanché, M. Bonamy, and N. Bonichon, Gallai's path decomposition in planar graphs. arXiv:2110.08870 (2021).
  • [BoPe19] Bonamy, Marthe and Perrett, Thomas J., *Gallai's path decomposition conjecture for graphs of small maximum degree*. Discrete Math. (2019), 1293--1299.
  • [CFZ26] Chu, Yanan and Fan, Genghua and Zhou, Chuixiang, *Gallai's conjecture and the path number of odd semi-cliques*. Discrete Math. (2026), Paper No. 114725, 6.
  • [Ch78] Chung, F. R. K., On partitions of graphs into trees. Discrete Math. (1978), 23-30.
  • [DeKo00] Dean, Nathaniel and Kouider, Mekkia, Gallai's conjecture for disconnected graphs. Discrete Math. (2000), 43--54.
  • [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
  • [Fa02] Fan, Genghua, Subgraph coverings and edge switchings. J. Combin. Theory Ser. B (2002), 54-83.
  • [Lo68] Lovász, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.
  • [Py96] Pyber, L., Covering the edges of a connected graph by paths. J. Combin. Theory Ser. B (1996), 152-159.

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.