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

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.