An odd graceful labeling of a graph G with q edges is an injection f from  to  such that when an edge xy is assigned the label  then the resulting edge labels are  A graph H is called a supersubdivision graph of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph  (m may vary for each edge) and identifying u and v with the two vertices in  that form one of the two partite sets. In this paper, we prove that the graphs obtained by the arbitrary supersubdivision of path, caterpillar and shell graphs are odd graceful.

arbitrary supersubdivision, odd graceful labeling, path, caterpillar, shell graph.

Received: May 30, 2022; Accepted: July 15, 2022; Published: September 27, 2022

How to cite this article: A. Velankanni, A. Bernick Raj and M. Sujasree, Odd graceful labeling of arbitrary supersubdivision of certain graphs, Advances and Applications in Discrete Mathematics 34 (2022), 23-37.  http://dx.doi.org/10.17654/0974165822041

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References
[1] Anita Pasotti, On d-graceful labelings, Ars Combinatoria 111 (2013), 207-223.
[2] C. Barrientos, Odd-graceful labelings of trees of diameter 5, AKCE International Journal of Graphs and Combinatorics 6 (2009), 307-313.
[3] A. Gallian Joseph, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics (2019), #DS6.
[4] R. B. Gnanajothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, 1991.
[5] S. W. Golomb, How to Number a Graph, Graph Theory and Computing, R. C. Reed, ed., Academic Press, New York, 1972, pp. 23-37.
[6] J. Jeba Jesintha and K. Ezhilarasi Hilda, Subdivided shell flower graphs are odd graceful, International Journal of Innovation in Science and Mathematics 2(4) (2014), 402-403.
[7] A. Rosa, On Certain Valuation of the Vertices of a Graph, Theory of Graphs, International Symposium, Rome, July 1966, Dunod, Paris, 1967, pp. 349-355.
[8] M. A. Seoud, A. T. Diab and E. A. Elsahawi, On strongly-C harmonious relatively prime, odd graceful and cordial graphs, Proc. Math. Phys. Soc., Egypt 73 (1998), 33-55.
[9] G. Sethuraman and P. Selvaraju, Decomposition of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs, Discrete Mathematics 260(1-3) (2003), 137-149.
[10] S. K. Vaidya and N. H. Shah, Graceful and odd graceful labeling of some graphs, International Journal of Mathematics and Soft Computing 3(1) (2013), 61-68.
[11] Q. T. Yan, Odd gracefulness and odd strongly harmoniousness of the product graphs J. Systems Sci. Math. Sci. 30 (2010), 341-348.