International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 122-127

# A $$\varGamma$$-magic Rectangle Set and Group Distance Magic Labeling

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)
3. 3.
Cichacz, S.: Note on group distance magic complete bipartite graphs. Cent. Eur. J. Math. 12(3), 529–533 (2014)
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)
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)
8. 8.
Froncek, D.: Group distance magic labeling of Cartesian product of cycles. Australas. J. Combin. 55, 167–174 (2013)
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)
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)
12. 12.
Harmuth, T.: Ueber magische Rechtecke mit ungeraden Seitenzahlen. Arch. Math. Phys. 66, 413–447 (1881)
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 