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A note on the total domination number. (English) Zbl 1161.05056
Summary: Let $$\gamma_t(G)$$ be the total domination number of a graph $$G$$ and $$G\square H$$ be the Cartesian product of graphs $$G$$ and $$H$$. For any graphs $$G$$, $$H$$ without isolated vertices, Henning and Rall show that $$\gamma_t(G)\gamma_t(H)\leq 6\gamma_t(G\square H)$$. In this note, we show that $$\gamma_t(G)\gamma_t(H)\leq 2\gamma_t(G\square H)$$ which answers the question in [M. A. Henning and D. F. Rall, Graphs Comb. 21, No. 1, 63–69 (2005; Zbl 1062.05109)]. In addition, we provide some examples to show that the inequality is sharp.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
total domination number