A note on the total domination number - Ho, Pak Tung (Utilitas Mathematica, 2008)
Let γt(G) be the total domination number of a graph G and G□H be the Cartesian product of graphs G and H. For any graphs G, H without isolated vertices, Henning and Rail show that γt(G) γt(H) ≤ 6γt(G□H). In this note, we show that γt(G)γt(H) ≤ 2 γt(G□H) which answer the question in [3]. In addition, we provide some examples to show that the inequality is sharp.