Source labels openChecked July 26, 2026

Ringel's Conjecture

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

Mathematical statement

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 $T.\text{map}(f_i)$ of $T$ under a vertex embedding $f_i : V \hookrightarrow \text{Fin}(2n+1)$ ; the copies are pairwise edge-disjoint and together cover every edge of $K_{2n+1}$ .

Statement source: Papers statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

ringel_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

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))) := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization