Upper bounds on the signed total domatic number of graphs

• Authors: Lutz Volkmann
• Venue: Discrete Applied Mathematics
• Year: 2011

Let G be a graph with the vertex set V(G), and let f:V(G)-{-1,1} be a two-valued function. If G has no isolated vertices and @?"x"@?"N"("v")f(x)=1 for each v@?V(G), where N(v) is the neighborhood of v, then f is a signed total dominating function on G. A set {f"1,f"2,...,f"d} of signed total dominating functions on G with the property that @?"i"="1^df"i(x)@?1 for each x@?V(G) is called a signed total dominating family (of functions) on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number of G, denoted by d"t^S(G). In this article we mainly present upper bounds on d"t^S(G), in particular for regular graphs. As an application of these bounds, we show that d"t^S(G)@?n-3 for any graph G of order n=4 without isolated vertices. Furthermore, we prove the Nordhaus-Gaddum inequality d"t^S(G)+d"t^S(G@?)@?n-3 for graphs G and G@? of order n=7 without isolated vertices, where G@? is the complement of G.