On the Signed Total Domatic Number of a Graph

Abstract

In this paper, we define the signed total domatic number of a graph in an analogous way to that of the fractional domatic number defined by Rall (A fractional version of domatic number. Congr. Numer. \(74 (1990), 100-106)\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. A set \(\{f_1,\ldots,f_a\}\) of signed total dominating functions on \(G\) such that \(\sum\limits_{i=1}^a f_i(v) \leq 1\) for each vertex \(v \in V(G)\) is called a signed total dominating family of functions on \(G\). The signed total domatic number of \(G\) is the maximum number of functions in a signed total dominating family of \(G\). In this paper, we investigate the signed total domatic number for special classes of graphs.