Ringel's Conjecture — open problem in graph theory

Ringel's Conjecture

For any tree T with n edges, the complete graph K_2n+1 decomposes into 2n+1 edge-disjoint copies of T. A "copy" of T is the image T.map(f_i) of T under a vertex embedding f_i : V hookrightarrow Fin(2n+1); the copies are pairwise edge-disjoint and together cover every edge of K_2n+1.

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-ringel-conjecture,
  title        = {Ringel's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-ringel-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

For any tree $T$ with $n$ edges, the complete graph $K_{2 n + 1}$ decomposes into $2 n + 1$ edge-disjoint copies of $T$ .

A "copy" of $T$ is the image $T . map (f_{i})$ of $T$ under a vertex embedding $f_{i} : V ↪ Fin (2 n + 1)$ ; the copies are pairwise edge-disjoint and together cover every edge of $K_{2 n + 1}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.RingelConjecture.

theorem ringel_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)),
      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

G. Ringel, Problem 25, in Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Academia, Prague, 1964.

Ringel's conjecture (1963): the complete graph $K_{2 n + 1}$ decomposes into copies of any tree with $n$ edges. It remains open; the case of all sufficiently large $n$ 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.