Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 295–301 | Cite as

On (a, d )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

  • M. F. Semeniuta


The results for the corona P n  ∘ P1 are generalized, which make it possible to state that P n  ∘ P1 is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q n is obtained. Functional dependencies are found that generate this type of labeling for Q n . It is proved by the method of mathematical induction that Q n is a (2 n  + n − 1, n − 2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G ∪ G is established.


distance magic labeling ( a, d) -distance antimagic labeling 1-vertex bimagic vertex labeling n-dimensional cube corona 


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  1. 1.
    J. A. Gallian, “A dynamic survey of graph labeling,” The Electronic Journal of Combinatorics. 2016. DS6: Dec 23.Google Scholar
  2. 2.
    B. M. Stewart, “Magic graphs,” Canad. J. Math., Vol. 18, 1031–1059 (1966).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. Stanley, “Linear homogeneous Diophantine equations and magic labelings of graphs,” Duke Math. J., Vol. 40, 607–632 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. S. Bloom and S. W. Golomb, “Numbered complete graphs, unusual rulers and assorted applications,” in: Theory and Applications of Graphs, Lecture Notes in Math., Vol. 642, Springer, New York (1978), pp. 53–53.Google Scholar
  5. 5.
    D. Froncek, “Fair incomplete tournaments with odd number of teams and large number of games,” Congressus Numerantium, Vol. 187, Iss. 1, 83–83 (2007).MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. A. Gallian (ed.), Mathematics and Sports: Mathematical Association of America (2010).Google Scholar
  7. 7.
    D. Froncek, “Handicap distance antimagic graphs and incomplete tournaments,” AKCE Int. J. Graphs Comb., Vol. 10, No. 2, 119–127 (2013).MathSciNetzbMATHGoogle Scholar
  8. 8.
    S. Arumugam and N. Kamatchi, “On (a; d)-distance antimagic graphs,” Australasian Journal of Combinatorics, Vol. 54, 279–287 (2012).Google Scholar
  9. 9.
    M. Nalliah, Antimagic Labelings of Graphs and Digraphs, Ph.D. Thesis, The National Centre for Advanced Research in Discrete Mathematics, University of Kalasalingam (2014).Google Scholar
  10. 10.
    M. F. Semeniuta, “On distance antimagic labeling of graphs,” in: Theory of Optimal Solutions, V. M. Glushkov Institute of Cybernetics of NASU (2016), pp. 26–32.Google Scholar
  11. 11.
    M. F. Semeniuta, “( , ) a d -distance antimagic labeling of some types of graphs,” Cybernetics and Systems Analysis, Vol. 52, No. 6, 950–955 (2016).Google Scholar
  12. 12.
    J. Baskar Babujee and S. Babitha, “On 1-vertex bimagic vertex labeling,” Tamkang Journal of Mathematics, Vol. 45, No. 3, 259–273 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Baskar Babujee, “Bimagic labeling in path graphs,” The Mathematics Education, Vol. 38, No. 1, 12–16 (2004).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Flight Academy of the National Aviation UniversityKropyvnytskyiUkraine

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