Odd Harmonious Labeling of Some Classes of Graphs

P. Jeyanthi; S. Philo

doi:10.4067/S0719-06462020000300299

DOI:

https://doi.org/10.4067/S0719-06462020000300299

Abstract

A graph \(G(p,q)\) is said to be odd harmonious if there exists an injection \(f: V(G)\rightarrow\left\{0, 1, 2,\cdots,2q-1\right\}\) such that the induced function \(f^{*}: E(G)\rightarrow\left\{1, 3,\cdots,2q-1\right\}\) defined by \(f^{*}(uv) = f(u)+ f(v)\) is a bijection. In this paper we prove that \(T_p\)- tree, \(T\hat\circ P_m\), \(T\hat\circ 2P_m\), regular bamboo tree, \(C_n\hat\circ P_m\), \(C_n\hat\circ 2P_m\) and subdivided grid graphs are odd harmonious.

Keywords

harmonious labeling , odd harmonious labeling , transformed tree , subdivided grid graph , regular bamboo tree

P. Jeyanthi Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur - 628 215, Tamil Nadu, India.
S. Philo Department of Mathematics, St. Xavier‘s College, Palayamkottai, Tirunelveli -627002, Tamilnadu, India.

Pages: 299–314
Date Published: 2020-12-07
Vol. 22 No. 3 (2020)

J. A. Gallian, “A dynamic survey of graph labeling”, The Electronic Journal of Combinatorics, #DS6, 2019.

R. L. Graham and N. J. A. Sloane, “On additive bases and harmonious graphs”, SIAM J. Algebraic Discrete Methods, vol. 1, no. 4, pp. 382–404, 1980.

F. Harary, Graph theory, Addison-Wesley, Massachusetts, 1972.

S. M. Hegde and S. Shetty, “On graceful trees”, Appl. Math. E-Notes, vol. 2, pp. 192–197, 2002.

P. Jeyanthi, S.Philo and Kiki A. Sugeng, “Odd harmonious labeling of some new families of graphs”, SUT Journal of Mathematics, vol. 51, no. 2, pp. 181–193, 2015.

P. Jeyanthi and S. Philo, “Odd harmonious labeling of some cycle related graphs”, Proyecciones Journal of Mathematics, vol. 35, no. 1, pp. 85–98, 2016.

P. Jeyanthi and S. Philo, “Odd harmonious labeling of plus graphs”, Bull. Int. Math. Virtual Inst., vol. 7, no. 3, pp. 515–526, 2017.

P. Jeyanthi, S. Philo and M. K. Siddiqui, “Odd harmonious labeling of super subdivision graphs”, Proyecciones Journal of Mathematics, vol. 38, no. 1, pp. 1–11, 2019.

P. Jeyanthi and S. Philo, “Odd harmonious labeling of subdivided shell graphs”, International Journal of Computer Sciences and Engineering, vol. 7, no. 5, pp. 77–80, 2019.

P. Jeyanthi and S. Philo, “Odd harmonious labeling of certain graphs”, Journal of Applied Science and Computations, vol. 6 no. 4, pp. 1224–1232, 2019.

P. Jeyanthi and S. Philo, “Some results on odd harmonious labeling of graphs”, Bull. Int. Math. Virtual Inst., vol. 9, no. 3, pp. 567–576, 2019.

P. Jeyanthi and S. Philo, “Odd harmonious labeling of some new graphs”, Southeast Asian Bull. Math., vol. 43, no. 4, pp. 509–523, 2019.

P. Jeyanthi, S. Philo and M. Youssef, “Odd harmonious labeling of grid graph”, Proyecciones Journal of Mathematics, vol. 38, no. 3, pp. 411–429, 2019.

Z. H. Liang and Z. L. Bai, “On the odd harmonious graphs with applications”, J. Appl. Math. Comput., vol. 29, no. 1–2, pp. 105–116, 2009.

G. Pooranam, R. Vasuki and S. Suganthi, “Even vertex odd mean labeling of transformed trees”, SUT J. Math., vol. 52, no. 2, pp. 117–131, 2016.

V. Ramalakshmi, C. Sekar, “One modulo N gracefulness of regular bamboo tree and coconut tree”, International Journal on Applications of Graph Theory in wireless Ad Hoc Networks and Sensor Networks, vol. 6, no. 2, pp. 1–10, 2014.

P. Selvaraju, P. Balaganesan and J.Renuka, “Odd harmonious labeling of some path related graphs”, International J. of Math. Sci. and Engg. Appls., vol. 7, no. III, pp. 163–170, 2013.

S. K. Vaidya and N. H. Shah, “Some new odd harmonious graphs”, International Journal of Mathematics and Soft Computing, vol. 1, pp. 9–16, 2011.

S. K. Vaidya, N. H. Shah, “Odd harmonious labeling of some graphs”, International J.Math. Combin., vol. 3, pp. 105–112, 2012.