Control Systems and Computers, N3, 2019, Article 2

Control Systems and Computers, 2019, Issue 3 (281), pp. 15-22.

UDC 519.1

M.F. SEMENIUTA, PhD in Phys.-Math. Sciences, associate professor, Department of Physics and Mathematics Sciences of the Flight Academy of the National Aviation University,  st. Dobrovolsky, 1, Kropivnitsky, 25005, Ukraine,

Z.OSHERMAN, PhD in Phys.-Math. Sciences, senior lecturer, Department of Medical Physics and Information technology №2 of Donetsk National Medical University,
st. Pryvokzalnaya, 27, Liman, 84404, Donetsk region, Ukraine,


We have shown the connection between vertex labelings of magic graph and its overgraph. Also, we have introduced binary relation on the set of all -distance magic graphs, where , i=1, 2, … and proved, that it is equivalence relation. Therefore, we have explored the properties of the graphs, which are in this relation. Finally, we have offered the algorithm of constructing r-regular handicap graph  of order n, where 0(mod8), r  1,3(mod4) and 3 ≤ r n–5.

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Keywords: graph, D-distance magic labeling, (a, d)-distance antimagic labeling, handicap labeling, Ddistance matrix, equivalence relation, 1factor.

  1. Ringel, G., 1964. Problem 25 in Theory of Graphs and Its Application, Sympos. Smolenice, 1963, Nakl. CSAV, Praha, pp. 162.
  2. Rosa, A., 1967. “On certain valuations of the vertices of a graph. In: Theory of Graphs, Internat”, Symposium, Rome, July 1966. Gordon and Breach/Dunod, New York/Paris, pp. 349-355.
  3. Sedlacek, J., 1964. “Problem 27 in Theory of graphs and its applications”, Proc. Symposium Smolenice 1963, Nakl. CSAV, Praha, pp. 163-164.
  4. Stewart, B.M., 1996. “Supermagic complete graphs”, Canadian Journal of Mathematics, 19, pp. 427-438.
  5. Gallian, J.A., 2018. A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, DS6: Dec 21, 502 p.
  6. O’Neal, A., Slater, P., 2011. “An introduction to distance D magic graphs”, Journal of the Indonesian Mathematical Society. Special Edition, pp. 89-107.
  7. Arumugam, S., Kamatchi, N., 2012. “On (a, d)-distance antimagic graphs”, Australasian journal of combinatorics, 54, pp. 279-287.
  8. Froncek, D., 2013. “Handicap distance antimagic graphs and incomplete tournaments”, AKCE International Journal of Graphs and Combinatorics, 10 (2), pp. 119-127.
  9. Arumugam, S., Froncek, D., Kamatchi, N., 2011. “Distance magic graphs – a survey”, Journal of the Indonesian Mathematical Society, Special Edition, pp. 11-26.
  10. Anholcer, M., Cichacz, S., Peterin, I., 2016. “Spectra of graphs and closed distance magic labelings”, Discrete mathematics, 339, pp. 1915-1923.
  11. Arumugam, S., Kamatchi, N., 2014. “On the uniqueness of D-vertex magic constant”, Discussiones mathematicae graph theory, 34, pp. 279-286.
  12. Semeniuta, M., Shulhin, V., 2019. “Matrices associated with D-distance magic graphs and their properties”, Cybernetics and Systems Analysis, 55 (3), pp. 441-448.
  13. Semeniuta, M.F., Sherman, Z.A., Dmitriiev, O.N., 2018. “Incomplete tournaments and magic types of labeling”, Control Systems and Computers, 5, pp. 13-24. (In Russian)
  14. Shepanik, A., 2015. Graph labelings and tournament scheduling. MS Thesis. University of Minnesota Duluth, 55 p.
  15. Froncek, D., Shepanik, A., 2018. “Regular handicap graph of order nº0(mod8)”, Electronic Journal of Graph Theory and Applications, 6 (2), pp. 208-218.

Received 18.06.2019