Strong Transversals in Hypergraphs and Double Total Domination in Graphs

Let H be a 3-uniform hypergraph of order n and size m, and let T be a subset of vertices of H. The set T is a strong transversal in H if T contains at least two vertices from every edge of H. The strong transversal number τs(H) of H is the minimum size of a strong transversal in H. We show that 7τs(H) ≤ 4n+ 2m, and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph H of order n ≥ 6 and size m that achieves equality in this bound. A set S of vertices in a graph G is a double total dominating set of G if every vertex of G is adjacent to at least two vertices in S. The minimum cardinality of a double total dominating set of G is the double total domination number γ×2,t(G) of G. Let G be a connected graph of order n with minimum degree at least three. As an application of our hypergraph results, we show that γ×2,t(G) ≤ 6n/7 with equality if and only if G is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if G is not the Heawood graph, we show that γ×2,t(G) ≤ 11n/13, while if G is a cubic graph different from the Heawood graph, we show that γ×2,t(G) ≤ 5n/6, and this bound is sharp.