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Prime Labeling of Circular Ladder Graph (CLn)
Subject area: Science,Engineering and Technology · Area of research: Mathematics
Abstract
Prime labeling is an assignment of integers to the vertices which are relatively prime. The theory behind the graph labeling were introduced by Rosa in the 1960s. A huge number of prime labeling research works have been discovered for various types of graphs today after undergoing different methods. In this work, the prime labeling method has been introduced for the circular ladder graph ?CL?_n with 2n vertices when n is an even. The graph obtained by using cartesian product of C_n with n vertices and path graph of the form P_2 is called circular ladder graph and is denoted by ?CL?_n (i.e C_n?P_2=?CL?_n). We proved that ?CL?_n is a prime graph for two special cases when n is an even integer and n+1 is prime, and when n is an even integer and 2n+1 is prime. In addition, the cyclic vertex labeling method in the clockwise direction and anti-clockwise direction have been introduced. Moreover, proofs of the two theorems related to the above two cases are given. We used the theorem on ladder graphs in the work by A. H. Berliner et al. to prove these results.
Keywords
Circular Ladder Graph, Graph Labeling, Prime Labeling
References
[1] A. H. Berliner, N. Dean, J. Hook, A. Marr, A. Mbikika, and C. D. Mc Bee, Coprime and Prime Labeling of Graphs, Journal of Integer Sequence, Vol. 19, No. 2, (2016).
[2] D. M. T. B. Dissanayake, R. A. S. T. Abeysekara K. D. E. Dhananjaya, A. A. I. Perera and P. G. R. S. Ranasinghe, Prime Labeling of Complete Tripartite Graphs of the Form Kl,m,n,, Elixir International Journal, Vol. 132, pp. 53092-53094, (2019).
[3] M.D.M.C.P. Weerarathna, D.M.T.B. Dissanayake, W.V. Nishadi, K.D.E. Dhananjaya, T.R.D.S.M. Thennakoon, and A.A.I. Perera. Consecutive Cyclic k – Prime Labeling of a Ladder Graph and Its Application to Crab Graph, Iconic Research and Engineering Journal, Vol. 3, Issue 11, pp. 278-283, ISSN 2456 – 8880, ,(2020).
[4] S. Ashokkumar, and S. Maragathavali, Prime Labeling of Some Special Graphs, IOSR Journal of Mathematics (IOSR-JM), Vol. 11, Issue 1, Ver. 1, pp. 01-05, (Jan - Feb. 2015)
[5] S. Meena, and P. Kavitha, Some New Results on Prime Graphs. International Journal of Mathematics and Computer Science, Vol. 3, Issue 1, pp. 837-849, (2015).
[6] S. Sakthivel, and T. R. Karpagam, Prime Labeling of Some New Standard Graphs. Paripex - Indian Journal of Research, Vol. 6, Issue 2, pp. 343-345, (2017).
[7] T. R. D. S. M. Thennakoon, M. D. M. C. P. Weerarathna and A. A. I. Perera, Prime Labeling of Newly Constructed Graph Using Star Graphs and Complete Bipartite Graphs, Sumerianz Journal of Scientific Research, Vol. 3, No. 2, pp. 10-17,(2020).
[8] Tout, A. N. Dabboucy and K. Howalla, Prime Labeling of Graphs, Nat. Acad. Sci letters 11 pp 365-368,( 1982).
How to cite this paper
@article{1702650,
author = {A.P. Batuwita, M.D.M.C.P. Weerarathna, G.W.M.M.K. Dheerasinghe},
title = {Prime Labeling of Circular Ladder Graph (CLn)},
journal = {Iconic Research And Engineering Journals},
year = {2021},
volume = {4},
number = {10},
pages = {192-195},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1702650.pdf},
abstract = {Prime labeling is an assignment of integers to the vertices which are relatively prime. The theory behind the graph labeling were introduced by Rosa in the 1960s. A huge number of prime labeling research works have been discovered for various types of graphs today after undergoing different methods. In this work, the prime labeling method has been introduced for the circular ladder graph ?CL?_n with 2n vertices when n is an even. The graph obtained by using cartesian product of C_n with n vertices and path graph of the form P_2 is called circular ladder graph and is denoted by ?CL?_n (i.e C_n?P_2=?CL?_n). We proved that ?CL?_n is a prime graph for two special cases when n is an even integer and n+1 is prime, and when n is an even integer and 2n+1 is prime. In addition, the cyclic vertex labeling method in the clockwise direction and anti-clockwise direction have been introduced. Moreover, proofs of the two theorems related to the above two cases are given. We used the theorem on ladder graphs in the work by A. H. Berliner et al. to prove these results.},
keywords = {Circular Ladder Graph, Graph Labeling, Prime Labeling},
month = {April},
}