C&SA | Contents

Volume 56 >>> № 5 SEPTEMBER — OCTOBER 2020

UDC 519.1

М.F. Semeniuta¹, G.A. Donets²

¹ Flight Academy of the National Aviation University, Kropyvnytskyi, Ukraine

² V.M. Glushkov Institute of Cybernetics, National Academy of Sciences
of Ukraine, Kyiv, Ukraine

ON GROUP LABELING OF SOME GRAPHS

Abstract. We analyze group labeling of magic and antimagic types. The relationship between them for graph and its complement is established. The concept of closed group distance magic labeling is introduced. The conditions for the existence of Z ^2m₂ -distance magic labeling of a graph C ^m₄ are found, and a method for its construction is proposed. The conditions for the existence of Z ^r₂ -distance magic and antimagic labelings of the Cartesian product of regular graphs are established. The results of group remote magic labeling of the connection of two graphs are obtained.

Keywords: D -distance magic labeling, group distance magic labeling, group distance antimagic labeling.

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REFERENCES

Sedlacek J. Problem 27. “Theory of graphs and its applications”. Proc. Symposium, Smolenice, 1963, Nakl. CSAV, Praha. 1964. P. 163–164.

Stewart B.M. Supermagic complete graphs. Canadian Journal of Mathematics. 1967. Vol. 19. P. 427–438.

Kotzig A., Rosa A. Magic valuations of finite graphs. Canadian Mathematical Bulletin. 1970. Vol. 13. P. 451–461.

Ringel G., Llado A.S. Another tree conjecture. Bulletin of the Institute of Combinatorics and its Applications. 1996. Vol. 18. P. 83–85.

Lih K.W. On magic and consecutive labelings of plane graphs. Utilitas Mathematica. 1983. Vol. 24. P. 165–197.

MacDougall J., Miller M., Wallis Slamin W.D. Vertex-magic total labelings. Utilitas Mathematica. 2002. Vol. 61. P. 3–21.

Gallian J.A. A dynamic survey of graph labeling. The Electronic Journal of Combinatorics. 2018. DS6: Dec 21. 502 p.

O’Neal A., Slater P. An introduction to distance D magic graphs. Journal of the Indonesian Mathematical Society. Special Edition. 2011. P. 89–107.

Froncek D. Group distance magic labeling of Cartesian product of cycles. Australasian Journal of Combinatorics. 2013. Vol. 55. P. 167–174.

Fukuchi Y. Graph labelings in elementary abelian groups. Discrete Mathematics. 1998. Vol. 189, Issues 1–3. P. 117–122.

Egawa Y. Graph labelings in elementary abelian 2-groups. Tokyo Journal of Mathematics. 1997. Vol. 20, N 2. P. 365–379.

Combe D., Nelson A.M., Palmer W.D. Magic labellings of graphs over finite abelian groups. Australasian Journal of Combinatorics. 2004. Vol. 29. P. 259–271.

Arumugam S., Kamatchi N. On -distance antimagic graphs. Australasian Journal of Combinatorics. 2012. Vol. 54. P. 279–287.

Semenyuta M.F. About (a,d)-distance antimagic and 1-vertex bimagic vertex markup of certain types of graphs. Kibernetika i sistemnyj analiz. 2018. Vol. 54, N 2. P. 134–141.

Cichacz S., Froncek D., Sugeng K., Zhou S. Group distance magic and antimagic graphs. Electronic Notes in Discrete Mathematics. 2015. Vol. 48. P. 41–48.

Harari F. Graph Theory [Russian translation]. Moscow: Mir, 1973. 304 p.

Cichacz S. Group distance magic labeling of some cycle-related graphs. Australasian Journal of Combinatorics. 2013. Vol. 57. P. 235–243.

Cichacz S., Froncek D. Distance magic circulant graphs. Discrete Mathematics. 2016. Vol. 339, Issue 1. P. 84–94.

Cichacz S., Dyrlaga P., Froncek D. Group distance magic Cartesian product of two cycles. Discrete Mathematics. 2020. Vol. 343, Iss 5. P. 1–12.