Geodesic
The Signed Total Roman k-Domatic Number Of A Graph
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1027-1038.

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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.
Keywords: signed total Roman k-dominating function, signed total Roman k-domination number, signed total Roman k-domatic number 
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Volkmann, Lutz. The Signed Total Roman k-Domatic Number Of A Graph. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 4, pp. 1027-1038. http://geodesic.mathdoc.fr/item/DMGT_2017_37_4_a12/

[1] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261. doi: 10.1002/net.3230070305

[2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).

[3] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177. doi: 10.2307/2306658

[4] P.J. Slater and E.L. Trees, Multi-fractional domination, J. Combin. Math. Combin. Comput. 40 (2002) 171-181.

[5] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016) 855-871. doi: 10.1007/s10878-015-9906-6

[6] L. Volkmann, Signed total Roman k-domination in graphs, J. Combin. Math. Com- bin. Comput., to appear.

[7] L. Volkmann, On the signed total Roman domination and domatic numbers of graphs, Discrete Appl. Math. 214 (2016) 179-186. doi: 10.1016/j.dam.2016.06.006

[8] C. Wang, The signed k-domination number in graphs, Ars Combin. 106 (2012) 205-211.