A note on the k-tuple total domination number of a graph

For every positive integer k and every graph G = (V, E) with minimum degree at least vertex set S is a k-tuple total domuinatin set (restp. k-tuple dominating set) of G, if for every vertex v is an element of V, vertical bar N-G(upsilon) boolean AND S vertical bar >= k, (resp. vertical bar N-G[upsilon] boolean AND S vertical bar >= k). The k-tuple total domination number 1,,t(C1) (resp. k-tuple domination number gamma xk,t (G) (resp. k-tuple domination number gamma xk (G) is the minimum cardinality of a k-tuple total dominating set (resp. k-tuple dominating set (resp. k-tuple dominating set ) of G. In this paper, we first prove that if to is a positive integer, then for which graphs G, gamma(xk,t)(G) = m or gamma(xk)(G) = m and give a necessary and sufficient condition for gamma(xk,t)(G) = gamma(x(k+1))(G). Then we show tint if C is a graph of order with delta(G) >= k + 1 >= 2, then gamma(xk,t)(G) has the lower bound 2 gamma(x(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.