On the upper total domination number of Cartesian products of graphs

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63-69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Gamma(t) (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Gamma(t) (G)Gamma(t) (H) <= 2 Gamma(t) (G square H).