reconstruction_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.

The reconstruction conjecture (Kelly 1941, Ulam 1960): Every finite simple graph on 3 or more vertices is uniquely determined (up to isomorphism) by its multiset of vertex-deleted subgraphs (the 'deck'). Proved for trees, regular graphs, and many other families.

Why this node was retired

The posted statement is

import Mathlib

theorem reconstruction_conjecture {V : Type*} [Fintype V] [DecidableEq V]
    (hV : 3 ≤ Fintype.card V)
    (G H : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel H.Adj]
    (hdeck : ∀ v : V, ∃ w : V,
      Nonempty ((G.induce (Set.univ \ {v})) ≃g (H.induce (Set.univ \ {w})))) :
    Nonempty (G ≃g H) := by
  sorry

Deck matches may reuse the same deleted vertex and ignore multiplicities.

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

Proposed corrected statement

Require a bijection σ of the vertex set such that G−v≅H−σ(v) for every v, or equality of the multisets of isomorphism classes of deleted graphs.

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.