Signed {k}-domatic numbers of graphs

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) - {±1,±2.....,±k} is called a signed {k}-dominating function if σuΕN(v) f(u)≥kfor each vertex vΕV(G). The signed {l}-dominating function is the same as the ordinary signed domination. A set {f1, f2,...,fd} of signed {k}-dominating functions on G with the property that σ di=1 fi(v) ≤ k: for each v Ε V(G), is called a signed {k}-dominating family (of functions) on G. The maximum number of functions in a signed {k}-dominating family on G is the signed {k}-domatic number of G, denoted by d{k}s(G). Note that d{l}s(G) is the classical signed domatic number ds(G). In this paper we initiate the study of signed {k}-domatic numbers in graphs, and we present some sharp upper bounds for d{k}s(G). In addition, we determine d {k}s(G) for several classes of graphs. Some of our results are extensions of known properties of the signed domatic number.