Total k-domination in Cartesian product of complete graphs.

Let G = (V, E) be a finite undirected graph. A set S of vertices in V is said to be total k-dominating if every vertex in V is adjacent to at least k vertices in S. The total k-domination number, gamma(kt)(G), is the minimum cardinality of a total k-dominating set in G. In this work we study the total k-domination number of Cartesian product of two complete graphs which is a lower bound of the total k-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total k-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. In particular, lim inf(n ->infinity) {gamma(kt)(G square H)/n : G, H are graphs of ordern} <= 2 ((sic)k/2(sic)(-1) + (sic)k+4/2(sic)(-1))(-1). We also prove that the equality is attained if k is even. The equality holds when G, H are both isomorphic to the complete graph, K-n, with n vertices. Furthermore, we obtain closed formulas for the total 2-domination number of Cartesian product of two complete graphs of whatever order. Besides, we prove that, for k = 3, the inequality above is improvable to lim inf(n ->infinity) gamma(3t)(K-n square K-n)/n <= 11/5. (c) 2023 Elsevier B.V. All rights reserved.