Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two

Abstract

We prove that every $4$-partite $4$-uniform hypergraph $H$ with matching number $\nu (H)=2$ satisfies $\tau (H)\le 6$, where $\tau$ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case $(r,\nu )=(4,2)$ of Ryser's conjecture. The best previous bound was $\tau \le 7$, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem ($\tau \le 3$ for intersecting $4$-partite $4$-uniform families), a short projection lemma (four base-disjoint edges in an intersecting family force a two-element cover), and Kőnig's matching theorem.

5 pages. Proves the (r,nu)=(4,2) case of Ryser's conjecture (Tuza's 1979 claim, recorded as open in DeBiasio et al. 2021). Every structural claim independently verified by exact computation (MILP + brute force). Proof found via four rounds of GPT-5.6 Pro; AI assistance disclosed in Methods