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 Gy\'arf\'as'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 K\H{o}nig's matching theorem.