Let H be a 3-regular 4-uniform hypergraph on n vertices. The
transversal number Ο(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 Ο(H)β€7n/18. Thomass\'{e} and Yeo [Combinatorica
27 (2007), 473--487] improved this bound and showed that Ο(H)β€8n/21.
We provide a further improvement and prove that Ο(H)β€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 Ξ(H)β€3, then Ο(H)β€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].Comment: 41 page