Signed total (j, k)-domatic numbers of graphs

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 distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ j for each x ∈ V (G), is called a signed total (j, k)-dominating family (of functions) on G, where j ≥ 1 is an integer. The maximum number of functions in a signed total (j, k)dominating family on G is the signed total (j, k)-domatic number of G, denoted by d (j,k) st (G). In this paper we initiate the study of the signed total (j, k)-domatic number. We present different bounds on d (j,k) st (G), and we determine the signed total (j, k)-domatic number for special graphs. Some of our results are extensions of well-known properties of different other signed total domatic numbers. 1 Terminology and introduction Various numerical invariants of graphs concerning domination were introduced by means of dominating functions and their variants (see, for example, Haynes, Hedetniemi and Slater [1, 2]). In this paper we define the signed total (j, k)-domatic number in an analogous way as Henning [4] has introduced the signed total domatic number. We consider finite, undirected and simple graphs G with vertex set V (G) = V and edge set E(G) = E. The cardinality of the vertex set of a graph G is called the order of G and is denoted by n(G) = n. If v ∈ V (G), then NG(v) = N(v) is the open neighborhood of v, i.e., the set of all vertices adjacent to v. The closed neighborhood NG[v] = N [v] of a vertex v consists of the vertex set N(v) ∪ {v}. The number dG(v) = d(v) = |N(v)| is the degree of the vertex v. The minimum and maximum degree of a graph G are denoted by δ(G) and Δ(G). The complement of a graph G is denoted by G. A fan is a graph obtained from a path by adding a new