The {k}-domatic number of a graph

Abstract

For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex $${v\in V(G)}$$ , the condition $${\sum_{u\in N[v]}f(u)\ge k}$$ is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f 1, f 2, . . . , f d } of {k}-dominating functions on G with the property that $${\sum_{i=1}^df_i(v)\le k}$$ for each $${v\in V(G)}$$ , is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d {k}(G). Note that d {1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d {k}(G). Many of the known bounds of d(G) are immediate consequences of our results.