GraphConjecture40 (WOWII #40): status note — equivalent deficiency form, bipartite case reduced to two finite lemmas, exhaustive verification to n = 11

Status note for FormalConjectures/WrittenOnTheWallII/GraphConjecture40.lean (research open). A preprint (https://doi.org/10.5281/zenodo.21778700; artifacts at https://github.com/cozuya/wowii-conjecture-40) establishes: (1) the conjecture is equivalent to ℓ + o ≥ 2τ + 1 for the decycling number τ = n − f, odd cycle transversal number o = n − b, and spanning-linear-forest edge count ℓ = n − p; (2) the bipartite case reduces to two explicitly stated open lemmas, with all surrounding structure (generalized Hall criterion, feasibility for every fixed edge, residual augmentation, descent) proved; (3) conditional on the bipartite case, ℓ + 2o ≥ 2τ + 1 holds for all connected nontrivial graphs; (4) the conjecture holds for every connected graph on ≤ 11 vertices (exhaustive, dual-engine, 0 violations; 1,006,700,565 graphs at n = 11). Posting for anyone working on this entry; the two open lemmas are stated precisely in §4 of the note and are plausibly formalizable targets themselves.

Read FormalConjectures/WrittenOnTheWallII/GraphConjecture40.lean and the preprint’s §4, where the two open bipartite lemmas are stated. Determine whether either lemma can be formalized from the surrounding proved structure; done means a reviewed Lean formalization of a lemma or a clearly documented research result.

Written by the indexing model from the issue text.