Transversals and Independence in Linear Hypergraphs with Maximum Degree Two

For k >= 2, let H be a k-uniform hypergraph on n vertices and m edges. Let S be a set of vertices in a hypergraph H. The set S is a transversal if S intersects every edge of H, while the set S is strongly independent if no two vertices in S belong to a common edge. The transversal number, tau(H), of H is the minimum cardinality of a transversal in H, and the strong independence number of H, alpha(H), is the maximum cardinality of a strongly independent set in H. The hypergraph H is linear if every two distinct edges of H intersect in at most one vertex. Let H-k be the class of all connected, linear, k -uniform hypergraphs with maximum degree 2. It is known [European J. Combin. 36 (2014), 231-236] that if H E is an element of H-k, then (k +1)tau(H) <= n + m, and there are only two hypergraphs that achieve equality in the bound. In this paper, we prove a much more powerful result, and establish tight upper bounds on tau(H) and tight lower bounds on alpha(H) that are achieved for infinite families of hypergraphs. More precisely, if k >= 3 is odd and H is an element of H-k has n vertices and rn edges, then we prove that k(k(2) - 3)tau(H) <= (k - 2)(k +1)n+ (k - 1)(2)m+ k - 1 and k(k(2) - 3)alpha(H) >= (k(2) +k - 4)n -(k - 1)(2)m - (k - 1). Similar bounds are proven in the case when k >= 2 is even.