On the total k-domination number of graphs

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal number τk(H) of a hypergraph H is the minimum size of a subset S ⊆ V (H) such that |S ∩ e| ≥ k for every edge e ∈ E(H). We know that for any graph G of order n with minimum degree at least k, γ×k(G) ≤ γ×k,t(G) ≤ n. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for γ×k,t(G) < n. Then we characterize complete multipartite graphs G with γ×k(G) = γ×k,t(G). We also state that the total k-domination number of a graph is the k-transversal number of its open neighborhood hypergraph, and also the domination number of a graph is the transversal number of its closed neighborhood hypergraph. Finally, we give an upper bound for the total k-domination number of the cross product graph G × H of two graphs G and H in terms on the similar numbers of G and H. Also, we show that this upper bound is strict for some graphs, when k = 1.