Signed star 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 f : E(G) → {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k 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 ∑d i=1 fi(e) ≤ 1 for each e ∈ E(G), is called a signed star k-dominating family (of functions) on G. The maximum number of functions in a signed star k-dominating family on G is the signed star k-domatic number of G, denoted by dkSS(G). In this paper we study the properties of the signed star k-domatic number dkSS(G). In particular, we determine the signed star k-domatic number of some classes of graphs. Some of our results extend these one given by Atapour et al. [1] for the signed star domatic number.