Transversal Game on Hypergraphs and the 3/4-Conjecture on the Total Domination Game

The 34 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G) ≤ 34n, where γtg(G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τg(H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H C4, then τg(H) ≤ 4 11 (nH + mH ), where nH and mH denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γtg(G) < 8 11n. As a consequence of this result, the 34 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.