New bounds on a generalization of Tuza's conjecture

Alex Parker¹

¹Iowa State University

The electronic journal of combinatorics, Tome 32 (2025) no. 2

Résumé

For a $k$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set $S$ of edges of $H$ whose pairwise intersection has size less than $m$. Let $\tau^{(m)}(H)$ denote the minimum size of a set $S$ of $m$-sets of $V(H)$ such that every edge of $H$ contains some $m$-set from $S$. A conjecture by Aharoni and Zerbib, which generalizes a conjecture of Tuza on the size of minimum edge covers of triangles of a graph, states that for a $k$-uniform hypergraph $H$, $\tau^{(k - 1)}(H)/\nu^{(k - 1)}(H) \leq \left \lceil \frac{k + 1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's conjecture holds when $\nu^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a graph class which satisfies Tuza's conjecture. We also prove various bounds on $\tau^{(m)}(H)/\nu^{(m)}(H)$ for other values of $m$ as well as some bounds on the fractional analogues of these numbers.

Détail

DOI : 10.37236/13355

Classification : 05C65, 05C70
Mots-clés : 3-uniform hypergraphs, covers, matchings