New bounds on a generalization of Tuza's conjecture
Alex Parker
TL;DR
The paper advances a generalized Tuza framework for k-uniform hypergraphs by bounding the ratio τ^{(m)}(H)/ν^{(m)}(H) for key cases and exploring fractional variants. Its main result shows the Tuza-type bound h(k,k−1) ≤ ⌈(k+1)/2⌉ holds when the (k−1)-matching number is at most 3, with corollaries for graph classes where Tuza’s conjecture holds. It provides sharp or near-sharp bounds for g_1(k,m) across several ranges of m, including m close to k and m around k/2, and extends these results to fractional versions, yielding concrete upper bounds like g_1^*(2k,k) and h^*(k,m). The methods combine structural analyses of maximum (k−1)-matchings, decomposition into connected/disconnected configurations, and fractional LP techniques, contributing both exact and asymptotic insights with potential graph-theoretic applications. These bounds improve understanding of covers and matchings in hypergraphs and illuminate cases where Tuza’s conjecture extends to broader combinatorial settings.
Abstract
For a $k$-uniform hypergraph $H$, let $ν^{(m)}(H)$ denote the maximum size of a set $S$ of edges of $H$ whose pairwise intersection has size less than $m$. Let $τ^{(m)}(H)$ 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 $H$, $τ^{(k - 1)}(H)/ν^{(k - 1)}(H) \leq \left \lceil \frac{k + 1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's conjecture holds when $ν^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a graph class which satisfies Tuza's conjecture. We also prove various bounds on $τ^{(m)}(H)/ν^{(m)}(H)$ for other values of $m$ as well as some bounds on the fractional analogues of these numbers.