Transversals in $4$-Uniform Hypergraphs

TL;DR

This paper improves bounds on the transversal number in 4-uniform hypergraphs, establishing the best possible upper limit and extending results to related graph domination parameters.

Contribution

It provides a tighter upper bound on the transversal number for 4-uniform hypergraphs with degree constraints, confirming a known conjecture and deriving a corollary for graph domination.

Findings

Proved that $ au(H) \\le 3n/8$ for 3-regular 4-uniform hypergraphs.

Extended the bound to hypergraphs with maximum degree 3, showing $ au(H) \\le n/4 + m/6$.

Derived an upper bound of $3n/7$ for the total domination number of graphs with minimum degree 4.

Abstract

Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990), 129--133] proved that $\tau(H) \le 7n/18$. Thomass\'{e} and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $\tau(H) \le 8n/21$. We provide a further improvement and prove that $\tau(H) \le 3n/8$, which is best possible due to a hypergraph of order eight. More generally, we show that if $H$ is a $4$-uniform hypergraph on $n$ vertices and $m$ edges with maximum degree $\Delta(H) \le 3$, then $\tau(H) \le n/4 + m/6$, which proves a known conjecture. We show that an easy corollary of our main result is that the total domination number of a graph on $n$ vertices with minimum degree at least~4 is at most $3n/7$, which was the main result of the Thomass\'{e}-Yeo paper [Combinatorica 27 (2007), 473--487].