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 ${\displaystyle {\gamma }_{\times k}\left(G\right)}$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, ${\displaystyle |N_G\left[v\right]\cap S|\ge k}$. Also the total k-domination number ${\displaystyle {\gamma }_{\times k,t}\left(G\right)}$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, ${\displaystyle |N_G\left(v\right)\cap S|\ge k}$. 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, ${\displaystyle {\gamma }_{\times k}\left(G\right)\le {\gamma }_{\times k,t}\left(G\right)\le n}$. Obviously for every k-regular graph, the upper bound n is sharp. Here, we give a sufficient condition for ${\displaystyle {\gamma }_{\times k,t}\left(G\right)<n}$. Then we characterize complete multipartite graphs G with ${\displaystyle {\gamma }_{\times k}\left(G\right)={\gamma }_{\times k,t}\left(G\right)}$. 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.