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 CichaczEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)


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.


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


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.AGH University of Science and TechnologyKrakówPoland

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