Open problemChecked July 24, 2026

Graceful Tree Conjecture

A universal labeling conjecture for finite trees.

Graph theory Combinatorics Graph labeling

Mathematical statement

Every tree with $m$ edges has an injective labeling $f:V\to\{0,\ldots,m\}$ whose edge differences are exactly $\{1,\ldots,m\}$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

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

GracefulLabeling.graceful_tree_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem graceful_tree_conjecture    {V : Type*} [Fintype V] [DecidableEq V]    (T : SimpleGraph V) [DecidableRel T.Adj]    (hT : T.IsTree) :    let m := T.edgeFinset.card    ∃ f : V → ℕ,      Function.Injective f ∧      (∀ v, f v ≤ m) ∧      T.edgeFinset.image (fun e =>        e.lift ⟨fun u v =>          Int.natAbs ((f u : ℤ) - (f v : ℤ)),          fun u v => by            rw [← Int.natAbs_neg, neg_sub]⟩) =        Finset.Icc 1 m := 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

Graceful labeling overviewbackground