Total domination in graphs with given girth

A set 5 of vertices in a graph G without isolated vertices is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. In this paper, we establish an upper bound on the total domination number of a graph with minimum degree at least two in terms of its order and girth. We prove that if G is a graph of order n with minimum degree at least two and girth g. then γ_t(G) ≤ n/2 + n/g, and this bound is sharp. Our proof is an interplay between graph theory and transversals in hypergraphs.