On the total domination number of cross products of graphs

Abstract

We give lower and upper bounds on the total domination number of the cross product of two graphs, ${\gamma }_t(G\times H)$. These bounds are in terms of the total domination number and the maximum degree of the factors and are best possible. We further investigate cross products involving paths and cycles. We determine the exact values of ${\gamma }_t(G\times \mathit{Pn})$ and ${\gamma }_t(C_n\times C_m)$ where $P_n$ and $C_n$ denote, respectively, a path and a cycle of length $n$.