Kotzig's Conjecture

formal-conjectures · paper · ams-5

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.

The $2n+1$ copies are $f_0, f_1, \dots, f_{2n}$ where $f_i(v) = f_0(v) + i$ for all vertices
$v$ — each copy is obtained by adding $i \pmod{2n+1}$ to every vertex of the base copy.
This is strictly stronger than RingelConjecture.ringel_conjecture.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Paper/KotzigConjecture.lean
Local target: Corpus.PaperKotzigConjecture.kotzig_conjecture
Source SHA-256: dd6a7c5aef9d2a381215ef22cd5b62bc3d07392e8c7a12307cf81d66dc55dfe1
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record