K-tuple total domatic number of a graph

Abstract

A set 5 of vertices in a graph G is a k-tuple total dominating set, abbreviated kTDS, of G if every vertex of G is adjacent to at least fc vertices in S. The minimum cardinality of a kTDS of G is the ktuple total domination number of G. For a graph to have a kTDS, its minimum degree is at least k. When k = 1, a k-tuple total domination number is the well-studied total domination number. The total domatic number of G has been denned as the largest number of sets in a partition of V into total dominating sets. Similarly, we define the fc-tuple total domatic number of G as the largest number of sets in a partition of V into k-tuple total dominating sets. We derive basic properties and bounds for the fc-tuple total domatic number. Many of the known bounds of total domatic number are immediate consequences of our results.