The Signed Total Roman k-Domatic Number Of A Graph

Volkmann, Lutz 

Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1027-1038

Résumé

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 Σ_ u ∈ N(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 f_1, f_2, . . ., f_d of distinct signed total Roman k-dominating functions on G with the property that Σ_i=1^d f_i(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 d_stR^k (G). In this paper we initiate the study of signed total Roman k-domatic numbers in graphs, and we present sharp bounds for d_stR^k (G). In particular, we derive some Nordhaus-Gaddum type inequalities. In addition, we determine the signed total Roman k-domatic number of some graphs.

Détail

Keywords: signed total Roman k-dominating function, signed total Roman k-domination number, signed total Roman k-domatic number