On the signed total Roman domination and domatic numbers of graphs

A signed total Roman dominating function on a graph G is a function f : V (G) -> {-1, 1, 2} such that Sigma(u is an element of N(v)) f (u) >= 1 for every vertex v is an element of V(G), where N(v) is the neighborhood of v, and every vertex u is an element of V(G) for which f (u) = -1 is adjacent to at least one vertex w for which f (w) = 2. The signed total Roman domination number of a graph G, denoted by gamma(stR)(G), equals the minimum weight of a signed total Roman dominating function. A set {f(1), f(2), . . .,f(d)} of distinct signed total Roman dominating functions on G with the property that Sigma(d)(i=1) f(i)(v) <= 1 for each v is an element of V(G), is called a signed total Roman dominating family (of functions) on G. The maximum number of functions in a signed total Roman dominating family on G is the signed total Roman domatic number of G, denoted by d(stR)(G). In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for gamma(stR)(G) and d(stR)(G). In addition, we determine the total signed Roman domatic number of some graphs. (C) 2016 Elsevier B.V. All rights reserved.