The (j, k)-domatic number of a graph

Abstract

Let k ≥ j ≥ 1 be two integers, and let G be a simple graph such that j(δ(G)+1) ≥ k, where δ(G) is the minimum degree of G. A (j, k)-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, …, j} such that for any vertex v ∊ V(G), the condition σu∊N[v] f(u) ≥ k is fulfilled, where N [v] is the closed neighborhood of v. A set {f1, f2, …, fd} of (j, k)-dominating functions on G with the property that σdi=1 fi(v ≤ j for each v ∊ V(G), is called a (j, k)-dominating family (of functions) on G. The maximum number of functions in a (j, k)-dominating family on G is the (j, k)-domatic number of G, denoted by d(j, k)(G). Note that d(1, 1)(G) is the classical domatic number d(G). In this paper we initiate the study of the (j, k)-domatic number in graphs and we present some bounds for d(j, k) (G). Many of the known bounds of d(G) are immediate consequences of our results.