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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.38 no.3 Antofagasta Aug. 2019 


Odd harmonious labeling of grid graphs

P. Jeyanthi1

S. Philo2 

Maged Z. Youssef3

1Govindammal Aditanar College for Women, Department of Mathematics, Research Centre, Tiruchendur - 628 215, Tamil Nadu, India

2Manonmaniam Sundaranar University. Research Scholar, Reg.No: 12193, Abishekappatti, Tirunelveli 627012, India. e-mail:

3Imam Mohammad Ibn Saud Islamic University, Department of Mathematics and Statistics, College of Science, Riyad11623, Saudi Arabia. e-mail:


A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, · · · , 2q − 1} such that the induced function f* : E(G) → {1, 3, · · · , 2q − 1} defined by f∗ (uv) = f (u) + f (v) is a bijection. In this paper we prove that path union of t copies of Pm×Pn, path union of t different copies of Pmᵢ×Pnᵢ where 1 ≤ i ≤ t, vertex union of t copies of Pm×Pn, vertex union of t different copies of Pmᵢ×Pnᵢ where 1 ≤ i ≤ t, one point union of path of Ptn (t.n.Pm×Pm), t super subdivision of grid graph Pm×Pn are odd harmonious graphs.

Keywords: Harmonious labeling; Odd harmonious labeling; Grid graph; Path union of graphs; One point union of path of graphs; t-super subdivision of graphs

Mathematics Subject Classification (2000):  05C78

Texto completo disponible sólo en PDF.

Full text available only in PDF format.


The authors thank the referee for the valuable comments to improve the presentation of the paper.


[1] J. Gallian, “A Dynamic Survey of Graph Labeling”, 20th ed. The Electronics Journal of Combinatorics, vol. # DS6, p. 432, 2007. [On line]. Available: ]

[2] R. Graham and N. Sloane, “On Additive Bases and Harmonious Graphs,” SIAM Journal on Algebraic Discrete Methods, vol. 1, no. 4, pp. 382-404, 1980, doi: 10.1137/0601045. [ Links ]

[3] F. Harary, Graph theory. Reading, MA: Addison-Wesley, 1972. [ Links ]

[4] Z. Liang and Z. Bai, “On the odd harmonious graphs with applications,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 105-116, Oct. 2008, doi: 10.1007/s12190-008-0101-0. [ Links ]

[5] P. Jeyanthi, S. Philo and K. Sugeng, “Odd harmonious labeling of some new families of graphs”, SUT Journal of Mathematics, vol. 51, no. 2, pp. 53-65, 2015. [On lime]. Available: Links ]

[6] P. Jeyanthi and S. Philo, “Odd harmonious labeling of some cycle related graphs,” Proyecciones (Antofagasta), vol. 35, no. 1, pp. 85-98, Mar. 2016, doi: 10.4067/S0716-09172016000100006. [ Links ]

[7] P. Jeyanthi and S. Philo, “Odd Harmonious Labeling of Plus Graphs”, Bulletin of the International Mathematical Virtual Institute , vol. 7, no. 3, pp. 515-526, Apr. 2017. [On line]. Available: ]

[8] P. Jeyanthi, S. Philo, and M. K. Siddiqui, “Odd harmonious labeling of super subdivisión graphs,” Proyecciones (Antofagasta) , vol. 38, no. 1, pp. 1-11, Feb. 2019, doi: 10.4067/S0716-09172019000100001. [ Links ]

[9] P. Jeyanthi and S. Philo, “Odd Harmonious Labeling of Some New Graphs”, Southeast Asian Bulletin of Mathematics, vol. 43, no. 4, pp. 509-523, 2019. [On line] Available: Links ]

[10] P. Jeyanthi and S. Philo, "Odd Harmonious Labeling of Subdivided Shell Graphs", International Journal of Computer Sciences and Engineering, vol. 7, no. 5 (Special), pp. 77-80, Mar. 2019, doi: 10.26438/ijcse/v7si5.7780. [ Links ]

[11] P. Jeyanthi and S. Philo, “Odd Harmonious Labeling of Certain Graphs”, Journal of Applied Science and Computations, vol. 6, no.4, pp. 1224-1232, Apr. 2019. [On line]. Available: ]

[12] P. Jeyanthi and S. Philo, “Some Results on Odd Harmonious Labeling”, Bulletin of the International Mathematical Virtual Institute, vol. 9, no. 3 pp. 567-576, May 2019. [On line]. Available: ]

[13] P. Selvaraju, P. Balaganesan and J. Renuka, “Odd Harmonious Labeling of Some Path Related Graphs”, International J. of Math. Sci. & Engg. Appls, vol. 7 no. 3, pp. 163-170, May 2013. [ Links ]

[14] S. Vaidya and N. Shah, “Some New Odd Harmonious Graphs”, International Journal of Mathematics and Soft Computing, vol. 1, no. 1 pp. 9-16, 2011. [On line]. Available: ]

[15] S. Vaidya and N. Shah , “Odd Harmonious Labeling of Some Graphs”, International J. Math. Combin, vol. 3, pp. 105-112, Sep. 2012. [On line]. Available: ]

Received: February 2018; Accepted: November 2018

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