The ( j , k ) - domatic number of a graph
نویسنده
چکیده
Let k ≥ j ≥ 1 be two integers, and letG 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 ∑ u∈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 ∑d i=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.
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