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Level A · Machine-checkable Hard Graph theory P-paper-kotzig-conjecture

Kotzig's Conjecture

For any tree T with n edges, the complete graph K_2n+1 decomposes into 2n+1 edge-disjoint copies of T via cyclic shifts of a single embedding.

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-paper-kotzig-conjecture,
  title        = {Kotzig's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-kotzig-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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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

For any tree with edges, the complete graph decomposes into edge-disjoint copies of via cyclic shifts of a single embedding.

The copies are where for all vertices — each copy is obtained by adding to every vertex of the base copy. This is strictly stronger than RingelConjecture.ringel_conjecture.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.KotzigConjecture.

theorem kotzig_conjecture {V : Type} [Finite V]
    (T : SimpleGraph V) (hT : T.IsTree)
    (n : ℕ) (hn : T.edgeSet.ncard = n) :
    ∃ f : Fin (2 * n + 1) → (V ↪ Fin (2 * n + 1)),
      (∀ i v, f i v = f 0 v + i) ∧
      Pairwise (fun i j => Disjoint (T.map (f i)).edgeSet (T.map (f j)).edgeSet) ∧
      ⨆ i, T.map (f i) = (⊤ : SimpleGraph (Fin (2 * n + 1)))

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

A. Kotzig, cited in A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, 1967, pp. 349–355.

Kotzig conjectured that for every , the complete graph decomposes into copies of any -edge tree via cyclic shifts of a single embedding. This is strictly stronger than Ringel's conjecture; see Paper/RingelConjecture.lean. The large- case is proved by Montgomery–Pokrovskiy–Sudakov; see Arxiv/2001.02665/RingelConjecture.lean.

Source and licence

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