Group distance magic Cartesian product of two cycles

Abstract: Let $G=(V,E)$ be a graph and $\Gamma $ an Abelian group both of order $n$. A $\Gamma$-distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow \Gamma $ for which there exists $\mu \in \Gamma $ such that $% \sum_{x\in N(v)}\ell (x)=\mu $ for all $v\in V$, where $N(v)$ is the neighborhood of $v$. Froncek %(\cite{ref_CicAus}) showed that the Cartesian product $C_m \square C_n$, $m, n\geq3$ is a $\mathbb{Z}{mn}$-distance magic graph if and only if $mn$ is even. It is also known that if $mn$ is even then $C_m \square C_n$ has $\mathbb{Z}{\alpha}\times \mathcal{A}$-magic labeling for any $\alpha \equiv 0 \pmod {{\rm lcm}(m,n)}$ and any Abelian group $\mathcal{A}$ of order $mn/\alpha$. %\cite{ref_CicAus} However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution this problem by proving some necessary conditions. We further prove that for $n$ even the graph $C_{n}\square C_{n}$ has a $\Gamma$-distance magic labeling for any Abelian group $\Gamma$ of order $n^{2}$. Moreover we show that if $m\neq n$, then there does not exist a $(\mathbb{Z}2)^{m+n}$-distance magic labeling of the Cartesian product $C{2^m} \square C_{2^{n}}$. We also give necessary and sufficient condition for $C_{m} \square C_{n}$ with $\gcd(m,n)=1$ to be $\Gamma$-distance magic.