Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs

Fery Firmansah, Muhammad Ridlo Yuwono


A graph G(p,q) with p=|V(G)| vertices and q=|E(G)| edges. The graph G(p,q) is said to be odd harmonious if there exist an injection f: V(G)->{0,1,2,...,2q-1} such that the induced function f*: E(G)->{1,2,3,...,2q-1} defined by  f*(uv)=f(u)+f(v) which is a bijection and f is said to be odd harmonious labeling of G(p,q). In this paper we prove that pleated of the Dutch windmill graphs C_4^(k)(r) with k>=1 and r>=1 are odd harmonious graph. Moreover, we also give odd harmonious labeling construction for the union pleated of the Dutch windmill graph   C_4^(k)(r) union C_4^(k)(r) with k>=1 and r>=1.


odd harmonious labeling, pleated graph, the Dutch windmill graph

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J. A. Gallian, "Dynamic Survey of Graph Labeling," The Electronics Journal of Combinatorics, vol. 19, pp. 1-408, 2016.

R. L. Graham and N. J. A. Sloane, "On Additive Bases and Harmonious Graphs," SIAM J. Algebra. Disc. Math, vol. 1, no. 4, pp. 382-404, 1980.

Z. H. Liang and Z. L. Bai, "On the Odd Harmonious Graphs with Applications," J Appl Math Comput, vol. 29, pp. 105-116, 2009.

F. Alyani, F. Firmansah, W. Giyarti and K. A. Sugeng, "The Odd Harmonious Labeling of KCn-Snake Graphs for Spesific Values of n, that is, for n=4 and n=8," in IndoMS International Conference on Mathematics and Its Applications IndoMS and UGM, Yogyakarta, 2013.

F. Firmansah, "Pelabelan Harmonis Ganjil pada Gabungan Graf Ular dan Graf Ular Berlipat," in Konferensi Nasional Matematika dan Pembelajarannya (KNPMP 1) UMS, Solo, 2016.

F. Firmansah and K. A. Sugeng, "Pelabelan Harmonis Ganjil pada Graf Kincir Angin Belanda dan Gabungan Graf Kincir Angin Belanda," Magistra, vol. XXVII, no. 94, pp. 56-92, 2015.

F. Firmansah and M. W. Syaifuddin, "Pelabelan Harmonis Ganjil pada Graf Kincir Angin Double Quadrilateral," in Seminar Nasioanal Matematika dan Pendidikan Matematika UNY, Yogyakarta, 2016.

M. E. Abdel-Aal, "Odd Harmonius Labeling of Cyclic Snakes," International Journal on Applications of Graph Theory in Wireless Adhoc Networks and Sensor Networks, vol. 5, no. 3, pp. 1-13, 2013.

M. E. Abdel-Aal, "New Families of Odd Harmonious Graphs," International Journal of Soft Computing, Mathematics, and Control, vol. 3, no. 1, pp. 1-13, 2014.

A. G. Saputri, K. A. Sugeng and D. Froncek, "The Odd Harmonious Labeling of Dumbell and Generalized Prism Graphs," AKCE Int. J. Graphs Comb, vol. 10, no. 2, pp. 221-228, 2013.

P. Jeyanthi and S. Philo, "Odd Harmonious Labeling of Some Cycle Related Graphs," Proyecciones Journal of Mathmatics, vol. 35, no. 1, pp. 85-98, 2016.

P. Jeyanthi, S. Philo and K. A. Sugeng, "Odd Harmonious Labeling of Some New Families of Graphs," SUT Journal of Mathematics, vol. 51, no. 2, pp. 53-65, 2015.

S. K. Vaidya and N. H. Shah, "Odd Harmonious Labeling of Some Graphs," International J. Math. Combin, vol. 3, pp. 105-112, 2012.

S. K. Vaidya and N. H. Shah, "Some New Odd Harmonious Graphs," International Journal of Mathematics and Soft Computing, vol. 1, no. 1, pp. 9-16, 2011.

DOI: http://dx.doi.org/10.18860/ca.v4i4.4043


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