The Signed Total Roman k-Domatic Number Of A Graph

Abstract

Let k ≥ 1 be an integer. A signed total Roman k-dominating function on a graph G is a function f : V (G) → {−1, 1, 2} such that Ʃu2N(v) f(u) ≥ k for every v ∈ V (G), where N(v) is the neighborhood of v, and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed total Roman k-dominating functions on G with the property that Ʃdi=1 fi(v) ≤ k for each v ∈ V (G), is called a signed total Roman k-dominating family (of functions) on G. The maximum number of functions in a signed total Roman k-dominating family on G is the signed total Roman k-domatic number of G, denoted by dkstR(G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for dkstR(G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.

Keywords

• signed total roman k-dominating function
• signed total roman k-domination number
• signed total roman k-domatic number.