hamiltonian_cycle_threshold

⚠️ 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.

Hamiltonian cycle threshold: In G(n,p), if p = (log n + log log n + c)/n then the probability of a Hamiltonian cycle approaches e^{-e^{-c}} as n→∞. This sharp threshold (Komlós–Szemerédi, Korshunov 1976) is related to the minimum degree condition. Exact threshold for Hamiltonicity in terms of minimum degree is an open direction.

Why this node was retired

The posted statement is

import Mathlib

theorem hamiltonian_cycle_threshold (n : ℕ) (hn : 3 ≤ n) (p : ℝ)
    (hp : (Real.log n + Real.log (Real.log n)) / n ≤ p ∧ p ≤ 1) :
    ∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
      (∀ v : Fin n, G.degree v ≥ 2) →
      ∃ f : ZMod n → Fin n,
        Function.Bijective f ∧
        ∀ i : ZMod n, G.Adj (f i) (f (i + 1)) := by
  sorry

A random-graph threshold becomes a universal statement about every graph of minimum degree2; p is unrelated to G.

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

Proposed corrected statement

Define G(n,p) as a random graph and state the asymptotic probability of Hamiltonicity for p=(log n+log log n+c)/n; do not infer Hamiltonicity of every graph with minimum degree2.

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.