  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].
