A note on the Tuza constant $c_k$ for small $k$

TL;DR

This paper investigates the Tuza constant $c_k$ for small $k$, providing improved bounds for certain values, enhancing understanding of hypergraph transversal ratios.

Contribution

It offers new upper and lower bounds on the Tuza constant $c_k$ for specific small values of $k$, improving previous results.

Findings

Upper bound on $c_k$ for $k extgreater=7$ is improved.

Lower bound on $c_k$ is improved for $k=7,8,10,11,13,14,17$.

Abstract

For a hypergraph $H$, the transversal is a subset of vertices whose intersection with every edge is nonempty. The cardinality of a minimum transversal is the transversal number of $H$, denoted by $\tau(H)$. The Tuza constant $c_k$ is defined as $\sup{\tau(H)/ (m+n)}$, where $H$ ranges over all $k$-uniform hypergraphs, with $m$ and $n$ being the number of edges and vertices, respectively. We give an upper bound and a lower bound on $c_k$. The upper bound improves the known ones for $k\geq 7$, and the lower bound improves the known ones for $k\in\{7, 8, 10, 11, 13, 14, 17\}$.