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

Abstract

Let γ ^{k} _t (G) denote the total {k}-domination number of graph G, and let $G\mathbin{\square}H$ denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, $\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)$ . As a corollary of this result, we have $\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)$ 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).