Signed total (k, k)-domatic number of a graph

Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number on G, denoted by dst(G). In this paper we initiate the study of the signed total (k, k)-domatic number, and we present different bounds on dst(G). Some of our results are extensions of known properties of the signed total domatic number dst(G) = d 1 st(G), given by Henning in 2006.