BOUNDS ON THE k-TUPLE DOMATIC NUMBER OF A GRAPH

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A vertex of a graph G dominates itself and all vertices adjacent to it. A subset S subset of V (G) is a k-tuple dominating set of G if each vertex of V (G) is dominated by at least k vertices in S. The k-tuple domatic number of G is the largest number of sets in a partition of V (G) into k-tuple dominating sets. In this paper, we present a lower bound on the k-tuple domatic number, and we establish Nordhaus-Gaddum inequalities. Some of our results extends those for the classical domatic number. (C) 2011 Mathematical Institute Slovak Academy of Sciences