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 $\sum _{e\in E(C)}f(e)\ge 1$ 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 $\sum _{i=1}^d{f}_i(e)\le 1$ 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 $d_{sc}^{\prime }(G)$. In this paper, we study the signed cycle domatic numbers in graphs and present sharp bounds for $d_{sc}^{\prime }(G)$. In addition, we determine the signed cycle domatic number of some special graphs.