On e-super (a,d)-edge antimagic total labeling of total graphs of paths and cycles

Document Type : Original paper

Authors

1 Department of Mathematics and Actuarial Science, B.S. Abdur Rahman Crescent Institute of Science and Technology, Chennai - 600048, Tamil Nadu, India

2 Department of Mathematic, D.B. Jain College, Chennai - 600097, Tamil Nadu, India

Abstract

A (p,q)-graph G is (a,d)-edge antimagic total if there exists a bijection f from V(G)∪E(G) to {1,2,…,p+q} such that for each edge uv∈E(G), the edge weight Λ(uv)=f(u)+f(uv)+f(v) forms an arithmetic progression with first term a>0 and common difference d≥0. An (a,d)-edge antimagic total labeling in which the vertex labels are 1,2,…,p and edge labels are p+1,p+2,…,p+q is called a {\it super} (a,d)-{\it edge antimagic total labeling}. Another variant of (a,d)-edge antimagic total labeling called as e-super (a,d)-edge antimagic total labeling in which the edge labels are 1,2,…,q and vertex labels are q+1,q+2,…,q+p. In this paper, we investigate the  existence of e-super (a,d)-edge antimagic total labeling for total graphs of paths, copies of cycles and disjoint union of cycles.

Keywords

Main Subjects

References

[1] M. Bača, E.T. Baskoro, and R. Simanjuntak, Super edge-antimagic labelings of the generalized Petersen graph P(n,⌊n−12⌋), Util. Math. 70 (2006), 119–127.
[2] M. Bača, Y. Lin, M. Miller, and M.Z. Youssef, Edge-antimagic graphs, Discrete Mathe. 307 (2007), no. 11-12, 1232–1244. https://doi.org/10.1016/j.disc.2005.10.038
[3] M. Bača, Y. Lin, and F.A. Muntaner-Batle, Super edge-antimagic labelings of the path-like trees, Util. Math. 73 (2007), 117–128.
[4] M. Bača, J. MacDougall, F. Bertault, M. Miller, R. Simanjuntak, and Slamin, Vertex-antimagic total labelings of graphs, Discuss. Math. Graph Theory 23 (2003), no. 1, 67–83.  http://doi.org/10.7151/dmgt.1186
[5] M. Bača and M. Miller, Super Edge-Antimagic Graphs: A Wealth of Problems and Some Solutions, Universal Publishers, 2008.
[6] R. Bodendiek and G. Walther, Arithmetisch antimagische graphen, graphentheorie III, Wagner and R. Bodendiek (eds.), Mannhein (1993).
[7] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Comb. 18 (2021), Article ID: #DS6.
[8] F. Harary, Graph Theory, Addison-Wesley Publishing Company, 1994.
[9] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, 1990.
[10] A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970), no. 4, 451–461.  https://doi.org/10.4153/CMB-1970-084-1
[11] Y. Lin, M. Miller, R. Simanjuntak, and Slamin, Magic and antimagic labelings of wheels, Proceedings of Eleventh Australasian Workshop of Combinatorial Algorithm (Australia) (J. Akiyama, E.T. Baskoro, and M. Kano, eds.), Hunter Valley,
2000.
[12] R. Simanjuntak, F. Bertault, and M. Miller, Two new (a,d)-antimagic graph labelings, Proceedings of Eleventh Australasian Workshop of Combinatorial Algorithm (Australia) (J. Ryan, ed.), Hunter Valley, 2000, pp. 179–189.
[13] K.A. Sugeng, M. Miller, and M. Barithm (Australia) (J. Akiyama, E.T. Baskoro, and M. Kano, eds.), Hunter Valley,
2000.
[14] K.A. Sugeng, M. Miller, and M. Barithm, Super antimagic total labeling of graphs, Util. Math. 76 (2008), 161–171.
[15] K.A. Sugeng, M. Miller, Slamin, and M. Bača, (a,d)-edge-antimagic total labelings of caterpillars, Combinatorial Geometry and Graph Theory (Berlin, Heidelberg) (J. Akiyama, E.T. Baskoro, and M. Kano, eds.), Springer Berlin Heidelberg,
2005, pp. 169–180. 
[16] K.A. Sugeng and W. Xie, Construction of super edge magic total graphs, Proceedings of Sixteenth Australasian Workshop of Combinatorial Algorithm (Australia) (J. Ryan, ed.), Ballarat, 2005, pp. 303–310.
Communications in Combinatorics and Optimization
Volume 10, Issue 4
December 2025
Pages 787-802

Files

Share

How to cite

Statistics