Journal of Applied Mathematics

On ( a , 1 ) -Vertex-Antimagic Edge Labeling of Regular Graphs

Martin Bača, Andrea Semaničová-Feňovčíková, Tao-Ming Wang, and Guang-Hui Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


An ( a , s ) -vertex-antimagic edge labeling (or an ( a , s ) -VAE labeling, for short) of G is a bijective mapping from the edge set E ( G ) of a graph G to the set of integers 1,2 , , | E ( G ) | with the property that the vertex-weights form an arithmetic sequence starting from a and having common difference s , where a and s are two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called ( a , s ) -antimagic if it admits an ( a , s ) -VAE labeling. In this paper, we investigate the existence of ( a , 1 ) -VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept ( a , s ) -vertex-antimagic edge deficiency, as an extension of ( a , s ) -VAE labeling, for measuring how close a graph is away from being an ( a , s ) -antimagic graph. Furthermore, the ( a , 1 ) -VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.

Article information

J. Appl. Math., Volume 2015 (2015), Article ID 320616, 7 pages.

First available in Project Euclid: 17 August 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Bača, Martin; Semaničová-Feňovčíková, Andrea; Wang, Tao-Ming; Zhang, Guang-Hui. On $(a,1)$ -Vertex-Antimagic Edge Labeling of Regular Graphs. J. Appl. Math. 2015 (2015), Article ID 320616, 7 pages. doi:10.1155/2015/320616.

Export citation


  • G. Chartrand and L. Lesniak, Graphs & Digraphs, Chapman & Hall/CRC Press, 1996.
  • J. A. Gallian, “A dynamic survey of graph labeling,” The Electronic Journal of Combinatorics, vol. 16, article DS6, 2013.
  • N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, San Diego, Calif, US, 1990.
  • N. Alon, G. Kaplan, A. Lev, Y. Roditty, and R. Yuster, “Dense graphs are antimagic,” Journal of Graph Theory, vol. 47, no. 4, pp. 297–309, 2004.
  • R. Bodendiek and G. Walther, “Arithmetisch antimagische Graphen,” in Graphentheorie III, K. Wagner and R. Bodendiek, Eds., BI-Wiss, Mannheim, Germany, 1993.
  • R. Bodendiek and G. Walther, “On (a, d)-antimagic parachutes,” Ars Combinatoria, vol. 42, pp. 129–149, 1996.
  • R. Bodendiek and G. Walther, “(a; d)-antimagic parachutes II,” Ars Combinatoria, vol. 46, pp. 33–63, 1997.
  • R. Bodendiek and G. Walther, “On arithmetic antimagic edge labelings of graphs,” Mitteilungen der Mathematischen Gesellschaft in Hamburg, vol. 17, pp. 85–99, 1998.
  • M. Bača and I. Holländer, “On (a; d)-antimagic prisms,” Ars Combinatoria, vol. 48, pp. 297–306, 1998.
  • M. Bača, “Antimagic labellings of antiprisms,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 35, pp. 217–224, 2000.
  • M. Miller, M. Bača, and Y. Lin, “On two conjectures concerning $(a;d)$-antimagic labelings of antiprisms,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 37, pp. 251–254, 2001.
  • M. Bača, “Consecutive-magic labeling of generalized Petersen graphs,” Utilitas Mathematica, vol. 58, pp. 237–241, 2000.
  • T. Nicholas, S. Somasundaram, and V. Vilfred, “On (a, d)- Antimagic special trees, unicyclic graphs and complete bipartite graphs,” Ars Combinatoria, vol. 70, pp. 207–220, 2004.
  • J. A. MacDougall, M. Miller, Slamin, and W. D. Wallis, “Vertex magic total labelings of graphs,” Utilitas Mathematica, vol. 61, pp. 3–21, 2002.
  • T.-M. Wang and G.-H. Zhang, “\emphE-super vertex magic regular graphs of odd degreečommentComment on ref. [19?]: Please update the information of this reference, if possible.,” Electronic Notes in Discrete Mathematics. In press.
  • J. Ivančo and A. Semaničová, “Some constructions of supermagic graphs using antimagic graphs,” SUT Journal of Mathematics, vol. 42, no. 2, pp. 177–186, 2006.
  • J. Petersen, “Die Theorie der regulären graphs,” Acta Mathematica, vol. 15, no. 1, pp. 193–220, 1891.
  • J. Holden, D. McQuillan, and J. M. McQuillan, “A conjecture on strong magic labelings of 2-regular graphs,” Discrete Mathematics, vol. 309, no. 12, pp. 4130–4136, 2009.
  • V. Swaminathan and P. Jeyanthi, “Super vertex magic labeling,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 6, pp. 935–939, 2003. \endinput