The signed (k, k)-domatic number of digraphs
نویسندگان
چکیده
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 x∈N−[v] f(x) ≥ k for each v ∈ 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 {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ∑d i=1 fi(x) ≤ k for each x ∈ 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 dS(D). In this paper, we initiate the study of the signed (k, k)-domatic number of digraphs, and we present different bounds on dS(D). Some of our results are extensions of well-known properties of the signed domatic number dS(D) = d 1 S(D) of digraphs D as well as the signed (k, k)-domatic number dS(G) of graphs G. AMS subject classifications: 05C20, 05C69, 05C45
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