hamiltonian_count_complete

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Number of Hamiltonian cycles in Kₙ = (n-1)!/2. Analogues for other specific graphs (bipartite Kₙ,ₙ, hypercubes, etc.) are open or only asymptotically known.

Why this node was retired

The posted statement is

import Mathlib

theorem hamiltonian_count_complete (n : ℕ) (hn : 3 ≤ n) :
    {f : ZMod n → Fin n | Function.Bijective f ∧
      ∀ i : ZMod n, (⊤ : SimpleGraph (Fin n)).Adj (f i) (f (i + 1))}.ncard =
    (n - 1).factorial / 2 := by
  sorry

The target counts parametrizations, retaining rotations and reversals, whereas the formula counts undirected cycles.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Count bijective cyclic parametrizations modulo the dihedral action of rotations and reversal; alternatively keep the raw function set and change its count to n!.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.