A famous conjecture of Tuza \cite{tuza} is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal number of edge-disjoint triangles. We propose a wider setting for this conjecture. For a hypergraph $H$ let $\nu^{(m)}(H)$ be the maximal size of a collection of edges, no two of which share $m$ or more vertices, and let $\tau^{(m)}(H)$ be the minimal size of a collection $C$ of sets of $m$ vertices, such that every edge in $H$ contains a set from $C$. We conjecture that the maximal ratio $\tau^{(m)}(H)/\nu^{(m)}(H)$ is attained in hypergraphs for which $\nu^{(m)}(H)=1$. This would imply, in particular, the following generalization of Tuza's conjecture: if $H$ is $3$-uniform, then $\tau^{(2)}(H)/\nu^{(2)}(H) \le 2$. (Tuza's conjecture is the case in which $H$ is the set of all triples of vertices of triangles in the graph). We show that most known results on Tuza's conjecture go over to this more general setting. We also prove some general results on the ratio $\tau^{(m)}(H)/\nu^{(m)}(H)$, and study the fractional versions and the case of $k$-partite hypergraphs.

中文翻译：

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图扎猜想的推广

Tuza \cite{tuza} 的一个著名猜想是覆盖图中所有三角形所需的最小边数最多是边不相交三角形最大数量的两倍。我们为这个猜想提出了一个更广泛的设置。对于超图 $H$，令 $\nu^{(m)}(H)$ 是边集合的最大尺寸，其中没有两条边共享 $m$ 或更多顶点，并令 $\tau^{( m)}(H)$ 是 $m$ 个顶点的集合 $C$ 的最小尺寸，这样 $H$ 中的每条边都包含来自 $C$ 的集合。我们推测最大比率 $\tau^{(m)}(H)/\nu^{(m)}(H)$ 是在 $\nu^{(m)}(H)= 的超图中获得的1美元。这将特别暗示 Tuza 猜想的以下推广：如果 $H$ 是 $3$-uniform，则 $\tau^{(2)}(H)/\nu^{(2)}(H) \乐 2$。（图萨' s 猜想的情况是 $H$ 是图中三角形顶点的所有三元组的集合）。我们表明，图扎猜想的大多数已知结果都适用于这种更一般的设置。我们还证明了比率 $\tau^{(m)}(H)/\nu^{(m)}(H)$ 的一些一般结果，并研究了分数版本和 $k$-partite 超图的情况.