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International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 122-127 | Cite as

A \(\varGamma \)-magic Rectangle Set and Group Distance Magic Labeling

  • Sylwia Cichacz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

Abstract

A \(\varGamma \)-distance magic labeling of a graph \(G = (V, E)\) with \(|V| = n\) is a bijection \(\ell \) from V to an Abelian group \(\varGamma \) of order n such that the weight \(w(x) =\sum _{y\in N_G(x)}\ell (y)\) of every vertex \(x \in V\) is equal to the same element \(\mu \in \varGamma \) called the magic constant. A graph G is called a group distance magic graph if there exists a \(\varGamma \)-distance magic labeling for every Abelian group \(\varGamma \) of order |V(G)|.

A \(\varGamma \)-magic rectangle set \(MRS_{\varGamma }(a, b; c)\) of order abc is a collection of c arrays \((a\times b)\) whose entries are elements of group \(\varGamma \), each appearing once, with all row sums in every rectangle equal to a constant \(\omega \in \varGamma \) and all column sums in every rectangle equal to a constant \(\delta \in \varGamma \).

In the paper we show that if a and b are both even then \(MRS_{\varGamma }(a, b; c)\) exists for any Abelian group \(\varGamma \) of order abc. Furthermore we use this result to construct group distance magic labeling for some families of graphs.

Keywords

Distance magic labeling Magic constant Sigma labeling Graph labeling Cartesian product \(\varGamma \)-magic rectangle set 

References

  1. 1.
    Barrientos, C., Cichacz, S., Froncek, D., Krop, E., Raridan, C.: Distance Magic Cartesian Product of Two Graphs (preprint)Google Scholar
  2. 2.
    Cichacz, S.: Group distance magic graphs \(G\times C_n\). Discrete Appl. Math. 177(20), 80–87 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cichacz, S.: Note on group distance magic complete bipartite graphs. Cent. Eur. J. Math. 12(3), 529–533 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cichacz, S., Froncek, D.: Distance magic circulant graphs. Preprint Nr MD 071 (2013). http://www.ii.uj.edu.pl/documents/12980385/26042491/MD_71.pdf
  5. 5.
    Combe, D., Nelson, A.M., Palmer, W.D.: Magic labellings of graphs over finite abelian groups. Australas. J. Comb. 29, 259–271 (2004)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory, Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Froncek, D.: Handicap distance antimagic graphs and incomplete tournaments. AKCE Int. J. Graphs Comb. 10(2), 119–127 (2013)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Froncek, D.: Group distance magic labeling of Cartesian product of cycles. Australas. J. Combin. 55, 167–174 (2013)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Sun, H., Yihui, W.: Note on magic squares and magic cubes on Abelian groups. J. Math. Res. Exposition 17(2), 176–178 (1997)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Comb. 17, 17–20 (2013). #DS6Google Scholar
  11. 11.
    Harmuth, T.: Ueber magische Quadrate undÉihnliche Zahlenfiguren. Arch. Math. Phys. 66, 286–313 (1881)zbMATHGoogle Scholar
  12. 12.
    Harmuth, T.: Ueber magische Rechtecke mit ungeraden Seitenzahlen. Arch. Math. Phys. 66, 413–447 (1881)zbMATHGoogle Scholar
  13. 13.
    Rao, S.B., Singh, T., Parameswaran, V.: Some sigma labelled graphs I. In: Arumugam, S., Acharya, B.D., Raoeds, S.B. (eds.) Graphs, Combinatorics, Algorithms and Applications, pp. 125–133. Narosa Publishing House, New Delhi (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakówPoland

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