Author’s abstract: We use the link between the existence of tilings in Manhattan metric with $\{1\}$-bowls and minimum total dominating sets of Cartesian products of paths and cycles. From the existence of such a tiling, we deduce the asymptotical values of total domination numbers of these graphs and we deduce the total domination numbers of some Cartesian products of cycles. Finally, we investigate the problem of total domination numbers for some Cartesian products of two paths.
Reviewer: Lutz Volkmann (Aachen)