Characterize group distance magic labeling of Cartesian product of two cycles

Abstract

Given a graph G with vertex set V(G) and an abelian group Γ of order |V(G)|, a Γ-distance magic labeling of G is a bijection φ:V(G)⟶Γ with the property that there exists μ∈Γ such that ∑y∈N(x)φ(y)=μ for any x∈V(G), where N(x) is the neighborhood of x and μ is the magic constant of this labeling.Froncek showed that the Cartesian product Cm□Cn admits a Zmn-distance magic labeling if and only if 2|mn, and asked for the full characterization of the group Γ such that Cm□Cn admits a Γ-distance magic labeling. Recently, Cichacz, Dyrlaga and Froncek made some progress toward this problem, but the general case is still open.In this paper, we completely solve this problem. We also determine all μ∈Zmn such that there exists a Zmn-distance magic labeling of Cm□Cn with magic constant μ.