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arXiv · 2609.14281

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

Patrick White
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Abstract

We prove that every $4$-partite $4$-uniform hypergraph $H$ with matching number $ν(H)=2$ satisfies $τ(H)\le 6$, where $τ$ 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 $(r,ν)=(4,2)$ of Ryser's conjecture. The best previous bound was $τ\le 7$, an integrality consequence of the theorem of Haxell and Scott (2012). The proof uses Gyárfás's intersecting-case theorem ($τ\le 3$ 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.

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Patrick White. 2026-09-13. Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two. https://arxiv.org/abs/2609.14281

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