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

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.

The author was supported by National Science Centre Grant Nr 2011/01/D/ ST1/04104.