# Zk-Magic Labeling of Open Star of Graphs

## Abstract

For any non-trivial abelian group A under addition a graph G issaid to be A-magic if there exists a labeling f : E(G) ! A − {0} such that, the vertex labeling f+ defined as f+(v) = Pf(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk, the group of integers modulo k and these graphs are referred as k-magic graphs. In this paper we prove that the graphs such as open star of shell, flower, double wheel, cylinder, wheel, generalised Petersen, lotus inside a circle and closed helm are Zk-magic graphs. Also we prove that

super subdivision of any graph is Zk-magic.

## References

[1] M. Doob. On the construction of the magic graphs, In F. Hoffman (Ed.), Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, Boca Raton, February 25-March l, 1974 (pp. 361–374), Florida Atlantic University 1974

[2] M. Doob. Generalizations of magic graphs, J. Combinatorial Theory, Series B, 17(3)(1974), 205–217.

[3] M. Doob. Characterizations of regular magic graphs, J. Combinatorial Theory, Series B, 25(1)(1978), 94–104.

[4] P. Jeyanthi and K. Jeya Daisy. Certain classes of Zk-magic graphs, J. Graph Labeling, (to appear).

[5] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of subdivision graphs, Discrete Math. Algorithm. Appl., 8(3)(2016), [19 pages] DOI: 10.1142/S1793830916500464.

[6] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of cycle of graphs, Journal of Algebra Combinatorics Discrete Structures and Applications, (to appear).

[7] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of some families of graphs, J. Algorithm. Comput., (to appear).

[8] J. A. Gallian. A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics, (2015), # DS6.

[9] V.J. Kaneria, M. Meghpara and H.M. Makadia. Graceful labeling for open star of graphs, Inter. J. Math. Stat. Invention, 2(9)(2014), 19–23.

[10] S.M. Lee, F. Saba, E. Salehi and H. Sun. On the V4- group magic graphs, Cong. Numer., 156(2002), 59–67.

[11] S.M. Lee, Hugo Sun and Lxin Wen. On group magic graphs, J. Combin. Math. Combin. Comput., 38(2001), 197-207.

[12] R.M. Low and S.M Lee. On group magic eulerian graphs, J. Combin. Math. Combin. Computing, 50(2004), 141-148.

[13] R.M. Low and S.M Lee. On the products of group-magic graphs, Australas. J. Combin., 34(2006), 41–48.

[14] J. Sedlacek. On magic graphs, Math. Slov., 26(4)(1976), 329-335.

[15] W. C. Shiu, PCB Lam and P.K. Sun. Construction of magic graphs and some A-magic graphs with A of even order, Congr. Numer., 167(2004), 97–107.

[16] R.P. Stanley. Linear homogeneous Diophantine equations and magic labelings of graphs, Duke Math. J., 40(3)(1973), 607–632.

[17] R.P. Stanley. Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Math. J., 43(3)(1976), 511–531.

[18] G.W. Sun and S.M. Lee. Construction of magic graphs, Cong. Numer., 103(1994), 243–251.

[2] M. Doob. Generalizations of magic graphs, J. Combinatorial Theory, Series B, 17(3)(1974), 205–217.

[3] M. Doob. Characterizations of regular magic graphs, J. Combinatorial Theory, Series B, 25(1)(1978), 94–104.

[4] P. Jeyanthi and K. Jeya Daisy. Certain classes of Zk-magic graphs, J. Graph Labeling, (to appear).

[5] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of subdivision graphs, Discrete Math. Algorithm. Appl., 8(3)(2016), [19 pages] DOI: 10.1142/S1793830916500464.

[6] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of cycle of graphs, Journal of Algebra Combinatorics Discrete Structures and Applications, (to appear).

[7] P. Jeyanthi and K. Jeya Daisy. Zk-magic labeling of some families of graphs, J. Algorithm. Comput., (to appear).

[8] J. A. Gallian. A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics, (2015), # DS6.

[9] V.J. Kaneria, M. Meghpara and H.M. Makadia. Graceful labeling for open star of graphs, Inter. J. Math. Stat. Invention, 2(9)(2014), 19–23.

[10] S.M. Lee, F. Saba, E. Salehi and H. Sun. On the V4- group magic graphs, Cong. Numer., 156(2002), 59–67.

[11] S.M. Lee, Hugo Sun and Lxin Wen. On group magic graphs, J. Combin. Math. Combin. Comput., 38(2001), 197-207.

[12] R.M. Low and S.M Lee. On group magic eulerian graphs, J. Combin. Math. Combin. Computing, 50(2004), 141-148.

[13] R.M. Low and S.M Lee. On the products of group-magic graphs, Australas. J. Combin., 34(2006), 41–48.

[14] J. Sedlacek. On magic graphs, Math. Slov., 26(4)(1976), 329-335.

[15] W. C. Shiu, PCB Lam and P.K. Sun. Construction of magic graphs and some A-magic graphs with A of even order, Congr. Numer., 167(2004), 97–107.

[16] R.P. Stanley. Linear homogeneous Diophantine equations and magic labelings of graphs, Duke Math. J., 40(3)(1973), 607–632.

[17] R.P. Stanley. Magic labelings of graphs, symmetric magic squares, systems of parameters and Cohen-Macaulay rings, Duke Math. J., 43(3)(1976), 511–531.

[18] G.W. Sun and S.M. Lee. Construction of magic graphs, Cong. Numer., 103(1994), 243–251.

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## Published

2016-12-27

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Чланци