A note on the total domination number

Abstract

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) ≤ 6_γt(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.