On total domination in the cartesian product of graphs

Abstract

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 K2 or Cn, Util. Math. 83 (2010) 313-322] by characterizing the pairs of graphs G and H for which γt(GH) = 12γt(G)γt(H), whenever γt(H) = 2. In addition, we present an infinite family of graphs Gn with γt(Gn) = 2n, which asymptotically approximate equality in γt(GnGn) ≥ 21γt(Gn)2