Total and paired domination numbers of toroidal meshes

Let G be a graph without isolated vertices. The total domination number of G is the minimum number of vertices that can dominate all vertices in G, and the paired domination number of G is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles Cn and Cm for any n≥ 3 and m ∈ {3,4}, and gives some upper bounds for n,m≥ 5.