SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A function f : V(G) -> {+/- 1, +/- 2,..., +/- k} is called a signed total {k}-dominating function if Sigma(u epsilon N) f(u) >= k for each vertex v epsilon V(G). A set {f(1), f(2),..., f(d)} of signed total {k}dominating functions on G with the property that Sigma(d)(i=1) f(i)(v) <= k for each v epsilon V (G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by d({k}S)(t) (G). Note that d({1}S)(t) (G) is the classical signed total domatic number d(S)(G). In this paper, we initiate the study of signed total k-domatic numbers in graphs, and we present some sharp upper bounds for d({k}S)(t) (G). In addition, we determine d({k}S)(t) (G) for several classes of graphs. Some of our results are extensions of known properties of the signed total domatic number.