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arXiv · 2609.35836

A Tight Fractional Version of Generalized Tuza's Conjecture

Sichen Wang
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Abstract

For an $r$-uniform hypergraph $H$, let $ν(H)$ be the maximum number of edges no two of which share $r-1$ vertices, and $τ(H)$ the minimum number of $(r-1)$-sets such that every edge contains one of them. Aharoni and Zerbib conjectured that $τ(H)\le\lceil\frac{r+1}{2}\rceil\,ν(H)$, which for $r=3$ generalizes Tuza's conjecture on triangles. We prove that the fractional relaxation $τ^*(H)$ of $τ(H)$ satisfies $τ^*(H)\le\frac{r+1}{2}\,ν(H)$. This constant is best possible for every $r$, and for $r\ge4$ it improves on the previous bound of roughly $3r/4$.

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Sichen Wang. 2026-09-24. A Tight Fractional Version of Generalized Tuza's Conjecture. https://arxiv.org/abs/2609.35836

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