Extensions and applications of the Tuza-Vestergaard theorem

Abstract

The transversal number $\tau \left(H\right)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if $H$ is a 3-regular 6-uniform hypergraph of order $n$, then $\tau \left(H\right)\le \frac{1}{4}n$. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph $H$ of order $n$ and size $m$. In particular, if $H$ is a 4-regular 6-uniform hypergraph of order $n$, then we show that $\tau \left(H\right)\le \frac{2}{7}n$. The Tuza constant $c_6$ is defined by $c_6=sup\frac{\tau \left(H\right)}{n\left(H\right)+m\left(H\right)}$, where the supremum is taken over the class of all 6-uniform hypergraphs $H$. Since 1990 the exact value of $c_6$ has yet to be determined. We show that $\frac{1}{6}\le c_6\le \frac{1}{6}+\frac{1}{210}$, where $c_6=\frac{1}{6}$ is conjectured to be the correct bound. Moreover we show that if $G$ is a graph of order $n$ with $\delta \left(G\right)\ge 6$, then ${\gamma }_t\left(G\right)\le \left(\frac{4}{13}+\frac{6}{217}\right)n$, where ${\gamma }_t\left(G\right)$ denotes the total domination number of $G$ and ${\gamma }_t\left(G\right)\le \frac{4}{13}n$ is conjectured to be the correct bound. These bounds improve best known bounds to date.