SIGNED TOTAL 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) ! f 1;1g is called a signed total k-dominating function if ∑ u2 N(v) f(u) k for each vertex v 2 V (G). A set ff1;f2;:::;fdg of signed total k-dominating functions of G with the property that ∑ d=1 fi(v) 1, for each v 2 V (G), is called a signed total k-dominating family (of functions) of G. The maximum number of functions in a signed total k-dominating family of G is the signed total k-domatic number of G, denoted by d t (G). In this note we initiate the study of the signed total k-domatic numbers of graphs and present some sharp upper bounds for this parameter. We also determine the signed total k-domatic numbers of complete graphs and complete bipartite graphs.