Linear hypergraphs with large transversal number and maximum degree two

For k>=2, let H be a k-uniform hypergraph on n vertices and m edges. The transversal number @t(H) of H is the minimum number of vertices that intersect every edge. Chvatal and McDiarmid [V. Chvatal, C. McDiarmid, Small transversals in hypergraphs, Combinatorica 12 (1992) 19-26] proved that @t(H)@?(n+@?k2@?m)/(@?3k2@?). In particular, for k@?{2,3} we have that (k+1)@t(H)@?n+m. A linear hypergraph is one in which every two distinct edges of H intersect in at most one vertex. In this paper, we consider the following question posed by Henning and Yeo: Is it true that if H is linear, then (k+1)@t(H)@?n+m holds for all k>=2? If k>=4 and we relax the linearity constraint, then this is not always true. We show that if @D(H)@?2, then (k+1)@t(H)@?n+m does hold for all k>=2 and we characterize the hypergraphs achieving equality in this bound.