Group distance magic Cartesian product of two cycles

Let G=(V,E) be a graph and Γ an Abelian group both of order n. A Γ-distance magic labeling of G is a bijection ℓ:V→Γ for which there exists μ∈Γ such that ∑_x∈N(v)ℓ(x)=μ for all v∈V, where N(v) is the neighborhood of v. Froncek showed that the Cartesian product C_m□C_n, m,n≥3 is a Z_mn-distance magic graph if and only if mn is even. It is also known that if mn is even then C_m□C_n has Z_α×A-magic labeling for any α≡0(modlcm(m,n)) and any Abelian group A of order mn∕α. 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 of this problem by proving some necessary conditions. We further prove that for n even the graph C_n□C_n has a Γ-distance magic labeling for any Abelian group Γ of order n². Moreover we show that if m≠n, then there does not exist a (Z₂)^m+n-distance magic labeling of the Cartesian product C_2^m □C_2ⁿ . We also give a necessary and sufficient condition for C_m□C_n with gcd(m,n)=1 to be Γ-distance magic.