tuza_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.

Tuza's conjecture (1981): For any graph G, if τ_Δ(G) is the minimum number of edges needed to cover all triangles and ν_Δ(G) is the maximum number of edge-disjoint triangles, then τ_Δ(G) ≤ 2·ν_Δ(G). Proved for planar graphs and many other special cases.

Why this node was retired

The posted statement is

import Mathlib

-- Tuza's conjecture (1981): if a graph G has no more than t edge-disjoint triangles,
-- then G has a set of at most 2t edges that intersects every triangle
-- Equivalent: the triangle covering number ≤ 2 × triangle packing number
theorem tuza_conjecture (V : Type*) [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (T : ℕ)
    (hpack : T = Finset.card (Finset.univ.filter (fun S : Finset V =>
      S.card = 3 ∧ ∀ v ∈ S, ∀ w ∈ S, v ≠ w → G.Adj v w))) :
    ∃ cover : Finset (Sym2 V),
      cover.card ≤ 2 * T ∧
      cover ⊆ G.edgeFinset ∧
      ∀ a b c : V, G.Adj a b → G.Adj b c → G.Adj a c →
        s(a, b) ∈ cover ∨ s(b, c) ∈ cover ∨ s(a, c) ∈ cover := by
  sorry

T is the total number of triangles, not the maximum size of an edge-disjoint triangle family.

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

Define ν as the maximal cardinality of a family of triangles with pairwise disjoint edge sets and require a triangle edge cover of size at most2ν.

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.