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Vertex-antimagic labelings of regular graphs

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Abstract

Let G = (V,E) be a finite, simple and undirected graph with p vertices and q edges. An (a, d)-vertex-antimagic total labeling of G is a bijection f from V (G) ∪ E(G) onto the set of consecutive integers 1, 2, …, p + q, such that the vertex-weights form an arithmetic progression with the initial term a and difference d, where the vertex-weight of x is the sum of the value f(x) assigned to the vertex x together with all values f(xy) assigned to edges xy incident to x. Such labeling is called super if the smallest possible labels appear on the vertices.

In this paper, we study the properties of such labelings and examine their existence for 2r-regular graphs when the difference d is 0, 1, …, r + 1.

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Authors and Affiliations

  1. College of Computer Science and Information Systems, Jazan University, Jazan, 45 142, Saudi Arabia

    Ali Ahmad

  2. Faculty of Mathematics, COMSATS Institute of Information Technology, Lahore Campus, 54000, Lahore, Pakistan

    Kashif Ali

  3. Department of Applied Mathematics and Informatics, Technical University, Letná 9, 04200, Košice, Slovakia

    Martin Bača & Andrea Semaničová-Feňovčíková

  4. Department of Applied Mathematics, VŠB-Technical University of Ostrava, 17. listopadu 15, 708 33, Ostrava-Poruba, Czech Republic

    Petr Kovář

Authors
  1. Ali Ahmad
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  2. Kashif Ali
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  3. Martin Bača
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  4. Petr Kovář
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  5. Andrea Semaničová-Feňovčíková
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Corresponding author

Correspondence to Martin Bača.

Additional information

Supported by Slovak VEGA Grant 1/0130/12, Higher Education Commission Pakistan (Grant No. HEC(FD)/2007/555) and by the Ministry of Education of the Czech Republic (Grant No. MSM6198910027)

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Ahmad, A., Ali, K., Bača, M. et al. Vertex-antimagic labelings of regular graphs. Acta. Math. Sin.-English Ser. 28, 1865–1874 (2012). https://doi.org/10.1007/s10114-012-1018-y

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  • DOI: https://doi.org/10.1007/s10114-012-1018-y

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