Transversals in $4$-Uniform Hypergraphs

Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $τ(H)$ 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 $τ(H) \le 7n/18$. Thomassé and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $τ(H) \le 8n/21$. We provide a further improvement and prove that $τ(H) \le 3n/8$, 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 $Δ(H) \le 3$, then $τ(H) \le n/4 + m/6$, 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 Thomassé-Yeo paper [Combinatorica 27 (2007), 473--487].

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