| Summary: | Ho proved in [A note on the total domination number, Util. Math. 77 (2008) 97–100] that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and K2or Cn, Util. Math. 83 (2010) 313–322] by characterizing the pairs of graphs G and H for which γt(G□H)=12γt(G)γt(H)$\gamma _t \left( {G\square H} \right) = {1 \over 2}\gamma _t \left( G \right)\gamma _t \left( H \right)$
, whenever γt(H) = 2. In addition, we present an infinite family of graphs Gn with γt(Gn) = 2n, which asymptotically approximate equality in γt(Gn□Hn)≥12γt(Gn)2$\gamma _t \left( {G_n \square H_n } \right) \ge {1 \over 2}\gamma _t \left( {G_n } \right)^2 $.
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