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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.

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)