Geodesic
Upper Bounds on the Signed Total (k, k)-Domatic Number of Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 641-650

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Let G be a 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 f_1, f_2, . . ., f_d of distinct signed total k-dominating functions on G with the property that ∑_i=1^d f_i(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed total (k, k)- domatic number, in particular for regular graphs.
Keywords: signed total (k, k)-domatic number, signed total k-dominating function, signed total k-domination number, regular graphs 
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     author = {Volkmann, Lutz},
     title = {Upper {Bounds} on the {Signed} {Total} (k, {k)-Domatic} {Number} of {Graphs}},
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Volkmann, Lutz. Upper Bounds on the Signed Total (k, k)-Domatic Number of Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 641-650. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a3/