Total k-domination in Cartesian product of complete graphs

Abstract

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, γkt(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 infn→∞γkt(G□H)n:G,H are graphs of ordern≤2k2−1+k+42−1−1. 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, Kn, 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 infn→∞γ3t(Kn□Kn)/n≤11/5.