Upper bounds on the k-tuple domination number and k-tuple total domination number of a graph

Given a positive integer k, a subset S of vertices of a graph G is called a k-tuple dominating set in G if for every vertex v is an element of V(G), vertical bar N[v] boolean AND S vertical bar >= k. The minimum cardinality of a k-tuple dominating set in G is the k-tuple domination number gamma(xk)(G) of G. A subset S of vertices of a graph G is called a k-tuple total dominating set in G if for every vertex v is an element of V(G), vertical bar N(v) boolean AND S vertical bar >= k. The minimum cardinality of a k-tuple total dominating set in G is the k-tuple total domination number gamma(xk,t)(G) of G. We present probabilistic upper bounds for the k-tuple domination number of a graph as well as for the k-tuple total domination number of a graph, and improve previous bounds given in [J. Harant and M.A. Henning, Discuss. Math. Graph Theory 25 (2005), 29-34], [E.J. Cockayne and A.G. Thomason, J. Combin. Math. Combin. Comput. 64 (2008), 251-254], and [M.A. Henning and A.P. Kazemi, Discrete Appl. Math. 158 (2010), 1006-1011] for graphs with sufficiently large minimum degree under certain assumptions.