Opuscula Math. 34, no. 3 (2014), 609-620
http://dx.doi.org/10.7494/OpMath.2014.34.3.609

Opuscula Mathematica

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

S. M. Sheikholeslami
L. Volkmann

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 $$d_{SS}^{(k,k)}(G)$$. 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.

Keywords: signed star $$(k,k)$$-domatic number, signed star domatic number, signed star $$k$$-dominating function, signed star dominating function, signed star $$k$$-domination number, signed star domination number, regular graphs.

Mathematics Subject Classification: 05C69.

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• S. M. Sheikholeslami
• Azarbaijan Shahid Madani University, Department of Mathematics, Research Group of Processing and Communication, Tabriz, I.R. Iran
• L. Volkmann
• RWTH-Aachen University, Lehrstuhl II für Mathematik, 52056 Aachen, Germany
• Revised: 2013-02-28.
• Accepted: 2014-01-23.