On the total {k}-domination number of Cartesian products of graphs

Let gamma({k})(t)(G) denote the total {k}-domination number of graph G, and let G square H denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, gamma({k})(t)(G)gamma({k})(t)(H) <= k(k + 1)gamma({k})(t)(G square H). As a corollary of this result, we have gamma(t)(G)gamma(t)(H) <= 2 gamma(t)(G square H) for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63-69, 2005).