On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem

Abstract

A $\Gamma$-magic rectangle set ${\mathrm{MRS}}_{\Gamma }(a,b;c)$ is a collection of $c$ arrays of size $a\times b$ whose entries are the elements of an abelian group $\Gamma$ of order $abc$, each one appearing once and in a unique array in such a way that the sum of the elements of each row is equal to a constant $\omega \in \Gamma$ and the sum of the elements of each column is equal to a constant $\delta \in \Gamma$. In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an ${\mathrm{MRS}}_{\Gamma }(a,b;c)$. We also generalize this problem, describing constructions of $\Gamma$-magic rectangle sets, whose elements are partially filled arrays.