A magic rectangle set on Abelian groups and its application

Abstract

A Γ-magic rectangle set MRSΓ(a,b;c) of order abc is a collection of c arrays (a×b) whose entries are elements of group Γ of order abc, each appearing once, with all row sums in every rectangle equal to a constant ω∈Γ and all column sums in every rectangle equal to a constant δ∈Γ.In this paper we prove that for {a,b}≠{2α,2k+1} where α and k>0 are some natural numbers, a Γ-magic rectangle set MRSΓ(a,b;c) exists if and only if a and b are both even or |Γ| is odd or Γ has more than one involution. We proved that there does not exist a Γ-magic rectangle set MRSΓ(2,2k+1;c) for any Abelian group Γ of order (4k+2)c. Moreover we obtain sufficient and necessary conditions for existence of a Γ-magic rectangle MRSΓ(a,b)=MRSΓ(a,b;1).This work will also be potentially useful for graph theorists who work in graph labeling. We apply the main result for construction of two group magic-type labelings for some families of graphs. In particular we give necessary and sufficient conditions for tKn,n to be Γ-magic.