k-tuple domatic in graphs

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination number. we define the k-tuple domatic number of g as the largest number of sets in a partition of v into k-tuple dominating sets. recall that when k = 1, a k-tuple domatic number is the well-studied domatic number. in this work, we derive basic properties and bounds for the k-tuple domatic number.