The $k$-tuple domatic number of a graph
Frank Harary, Teresa W. Haynes · Czech digital mathematics library · 1998
A node of a graph G = (V, E) dominates itself and all nodes adjacent to it.A subset S C V is a dominating set for G if each node is dominated by some node of S. This concept can be extended to k -tuple domination by requiring that each node in V be dominated by at least k nodes in S. The domatic number of G has been defined as the largest number of sets in a partition of V into dominating sets.Similarly, 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.We derive bounds for the k-tuple domatic number.Results involving the ordinary domination and domatic numbers are improved as a consequence of this generalized approach.