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 iota(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 iota(H) <= < ((n+ left perpendiculark/2right perpendicular m)/(left perpendicular3k/2right perpendicular) In particular, for k is an element of {2, 3} we have that (k + 1)iota(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)iota (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 Delta(H) <= 2, then (k + 1)iota (H) <= n + m does hold for all k >= 2 and we characterize the hypergraphs achieving equality in this bound. (C) 2013 Elsevier Ltd. All rights reserved.