caccetta_haggkvist_conjecture

⚠️ 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.

Caccetta–Häggkvist conjecture (1978): Every directed graph on n vertices with minimum out-degree ≥ ⌈n/3⌉ contains a directed 3-cycle. More generally, minimum out-degree ≥ n/k implies a directed cycle of length ≤ k. The k=3 case is the most studied; best known bound requires min out-degree ≥ 0.3n (Hamburger et al. 2007).

Why this node was retired

The posted statement is

import Mathlib

theorem caccetta_haggkvist_conjecture (n : ℕ) (hn : 1 ≤ n)
    (beats : Fin n → Fin n → Bool)
    (hmin_out : ∀ v : Fin n,
      n ≤ Nat.card {w : Fin n | beats v w = true} * n) :
    ∃ (cycle : Fin 3 → Fin n),
      Function.Injective cycle ∧
      beats (cycle 0) (cycle 1) = true ∧
      beats (cycle 1) (cycle 2) = true ∧
      beats (cycle 2) (cycle 0) = true := by
  sorry

The minimum degree inequality is n≤outdegreen rather than3outdegree≥n, and the conclusion insists on exactly3 distinct vertices.

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

Proposed corrected statement

For a loopless directed graph require3*outdegree(v)≥n and conclude a directed cycle of length at most3; if requesting exactly3, also exclude2-cycles and handle the intended oriented-graph setting.

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.