Signed star (j,k)-domatic number of a graph
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Abstract
Let $G$ be a simple graph without isolated vertices with edge set $E(G)$, and let $j$ and $k$ be two positive integers. A function $f\:E(G)\rightarrow \{-1, 1\}$ is said to be a signed star $j$-dominating function on $G$ if $\sum_{e\in E(v)}f(e)\ge j$ 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 distinct signed star $j$-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 $(j,k)$-dominating family (of functions) on $G$. The maximum number of functions in a signed star $(j,k)$-dominating family on $G$ is the signed star $(j,k)$-domatic number of $G$ denoted by $d^{(j,k)}_{SS}(G)$.In this paper we study properties of the signed star $(j,k)$-domatic number of a graph $G$. In particular, we determine bounds on $d_{SS}^{(j,k)}(G)$. Some of our results extend those ones given by Atapour, Sheikholeslami, Ghameslou and Volkmann [1] for the signed star domatic number, Sheikholeslami and Volkmann [5] for the signed star $(k,k)$-domatic number and Sheikholeslami and Volkmann [4] for the signed star $k$-domatic number.
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Sheikholeslami, S., & Volkmann, L.
(2014).
Signed star (j,k)-domatic number of a graph.
Acta Mathematica Universitatis Comenianae, 83(1), 19-28.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/72/20
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References
1. Atapour M., Sheikholeslami S. M., Ghameshlou A. N. and L. Volkmann, Signed star domatic number of a graph, Discrete Appl. Math., 158 (2010), 213-218.
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4. Sheikholeslami S. M. and Volkmann L., Signed star k-domatic number of a graph, Contrib. Discrete Math. 6 (2011), 20-31.
5. Signed star (k; k)-domatic number of a graph, submitted.
6. Wang C. P., The signed star domination numbers of the Cartesian product, Discrete Appl. Math. 155 (2007), 1497-1505.
7. The signed b-matchings and b-edge covers of strong product graphs, Contrib. Discrete Math. 5 (2010), 1-10.
8. Xu B., On edge domination numbers of graphs, Discrete Math. 294 (2005), 311-316.
9. Xu B., Two classes of edge domination in graphs, Discrete Appl. Math. 154 (2006), 1541-1546.
10. Xu B. and Li C. H., Signed star k-domination numbers of graphs, (Chinese) Pure Appl. Math. (Xi'an) 25 (2009), 638-641.
2. Haynes T. W., Hedetniemi S. T. and Slater P. J., Fundamentals of Domination in graphs, Marcel Dekker, Inc., New York, 1998.
3. Saei R. and Sheikholeslami S. M., Signed star k-subdomination numbers in graphs, Discrete Appl. Math. 156 (2008), 3066-3070.
4. Sheikholeslami S. M. and Volkmann L., Signed star k-domatic number of a graph, Contrib. Discrete Math. 6 (2011), 20-31.
5. Signed star (k; k)-domatic number of a graph, submitted.
6. Wang C. P., The signed star domination numbers of the Cartesian product, Discrete Appl. Math. 155 (2007), 1497-1505.
7. The signed b-matchings and b-edge covers of strong product graphs, Contrib. Discrete Math. 5 (2010), 1-10.
8. Xu B., On edge domination numbers of graphs, Discrete Math. 294 (2005), 311-316.
9. Xu B., Two classes of edge domination in graphs, Discrete Appl. Math. 154 (2006), 1541-1546.
10. Xu B. and Li C. H., Signed star k-domination numbers of graphs, (Chinese) Pure Appl. Math. (Xi'an) 25 (2009), 638-641.