The signed (k, k)-domatic number of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f : V (D) -> {-1, 1} be a two-valued function. If k >= 1 is an integer and Sigma(x is an element of N-[v]) f (x) >= k for each v is an element of V(D), where N- [v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f(1), f(2), ... , f(d)} of distinct signed k-dominating functions on D with the property that Sigma(d)(i=1) f(i)(x) <= k for each x is an element of V (D), is called a signed (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed (k, k)-dominating family on D is the signed (k, k)-domatic number on D, denoted by d(S)(k)(D). In this paper, we initiate the study of the signed (k, k)-domatic number of digraphs, and we present different bounds on d(S)(k)(D). Some of our results are extensions of well-known properties of the signed domatic number d(S)(D) = d(S)(1)(D) of digraphs D as well as the signed (k, k)-domatic number d(S)(k)(G) of graphs G.