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, τ(H), of H is the minimum cardinality of a transversal in H, and the strong independence number of H, α(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 Hk 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 ∈ Hk, then (k + 1)τ(H) 6 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 τ(H) and tight lower bounds on α(H) that are achieved for infinite families of hypergraphs. More precisely, if k > 3 is odd and H ∈ Hk has n vertices and m edges, then we prove that k(k2−3)τ(H) 6 (k−2)(k+1)n+(k−1)2m+k−1 and k(k2−3)α(H) > (k2 +k−4)n− (k−1)2m− (k−1). Similar bounds are proven in the case when k > 2 is even.