Signed star (k,k)-domatic number of a graph

Abstract

Let $G$ be a simple graph without isolated vertices with vertex set $V(G)$ and edge set $E(G)$ and let $k$ be a positive integer. A function $f:E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star $k$-dominating function on $G$ if $\sum_{e\in E(v)}f(e)\ge k$ for every vertex $v$ of $G$, where $E(v)=\{uv\in E(G)\mid u\in N(v)\}$. A set $\{f_1,f_2,\ldots,f_d\}$ of signed star $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(e)\le k$ for each $e\in E(G)$, is called a signed star $(k,k)$-dominating family (of functions) on $G$. The maximum number of functions in a signed star $(k,k)$-dominating family on $G$ is the signed star $(k,k)$-domatic number of $G$, denoted by $d^{(k,k)}_{SS}(G)$. In this paper we study properties of the signed star $(k,k)$-domatic number $d^{(k,k)}_{SS}(G)$. In particular, we present bounds on $d_{SS}^{(k,k)}(G)$, and we determine the signed $(k,k)$-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.