For every positive integer k and every graph G=(V,E) with minimum degree at least k, a vertex set S is a k-tuple total dominating set (resp. k -tuple dominating set) of G, if for every vertex vϵV, |N_G(v)∩ S|≥ k, (resp. |N_G[v]∩ S|≥ k). The k-tuple total domination number γ_{X k,t}(G) (resp. k-tuple domination number γ_{× k}(G)) is the minimum cardinality of a k-tuple total dominating set (resp. k-tuple dominating set) of G.
In this paper, we first prove that if m is a positive integer, then for which graphs G, γ_{× k,t}(G)=m or γ_{× k}(G)=m, and give a necessary and sufficient condition for γ_{× k,t}(G)=γ_{× (k+1)}(G). Then we show that if G is a graph of order n with δ(G)≥ k+1 ≥ 2, then γ_{× k,t}(G) has the lower bound 2γ_{× (k+1)}(G)-n and characterize graphs that equality holds for them. Finally we present two upper bounds for the k -tuple total domination number of a graph in terms of its order, minimum degree and k.

Tbilisi Mathematical Journal, Vol. 8(2) (2015), pp. 281-286