Upper bounds on the signed total domatic number of graphs

Abstract

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 d f 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 .