The Signed Cycle Domatic Number of a Graph

Abstract

For a nonempty graph \(G = (V(G), E(G))\), a signed cycle dominating function on \(G\) is introduced by Xu in 2009 as a function \(f : E(G) \to \{1, -1\}\) such that \(\sum_{e \in E(C)} f(e) \geq 1\) for any induced cycle \(C\) of \(G\). A set \(\{f_1, f_2, \dots, f_d\}\) of distinct signed cycle dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(e) \leq 1\) for each \(e \in E(G)\), is called a signed cycle dominating family (of functions) on \(G\). The maximum number of functions in a signed cycle dominating family on \(G\) is the signed cycle domatic number of \(G\), denoted by \(d’_{sc}(G)\). In this paper, we study the signed cycle domatic numbers in graphs and present sharp bounds for \(d’_{sc}(G)\). In addition, we determine the signed cycle domatic number of some special graphs.