chvatal_erdos_conjecture

⚠️ Retired — specification defect

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

Chvátal–Erdős conjecture (1972): Every graph G with vertex connectivity κ(G) ≥ α(G) (independence number) has a Hamiltonian cycle. Open in general; proved for planar graphs, tournaments, and specific graph families.

Why this node was retired

The posted statement is

import Mathlib

theorem chvatal_erdos_conjecture (V : Type*) [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (hconn : G.Connected)
    (hce : ∀ (kappa alpha : ℕ),
      (∀ S : Finset V, S.card < kappa →
        (G.induce (Set.univ \ S.toSet)).Connected) →
      (∀ I : Finset V,
        (∀ a ∈ I, ∀ b ∈ I, a ≠ b → ¬G.Adj a b) → I.card ≤ alpha) →
      kappa ≥ alpha) :
    ∃ f : ZMod (Fintype.card V) → V,
      Function.Bijective f ∧
      ∀ i : ZMod (Fintype.card V), G.Adj (f i) (f (i + 1)) := by
  sorry

The hypothesis universally ranges over all connectivity lower bounds and independence upper bounds rather than comparing the graph invariants.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Assume the graph is at least2-connected and its actual vertex connectivity κ(G) is at least its independence number α(G), with enough vertices for a cycle; conclude Hamiltonicity.

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.