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 τ ( H ) of H is the minimum number of vertices that intersect every edge. Chvátal and McDiarmid [V. Chvátal, C. McDiarmid, Small transversals in hypergraphs, Combinatorica 12 (1992) 19–26] proved that τ ( H ) ≤ ( n + ⌊ k 2 ⌋ m ) / ( ⌊ 3 k 2 ⌋ ) . In particular, for k ∈ { 2 , 3 } we have that ( k + 1 ) τ ( 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 ) τ ( 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 Δ ( H ) ≤ 2 , then ( k + 1 ) τ ( H ) ≤ n + m does hold for all k ≥ 2 and we characterize the hypergraphs achieving equality in this bound.