Group distance magic and antimagic hypercubes

Let G=(V,E) be a graph. A distance magic labeling of G is a bijective assignment of labels from {1,2,…,|V(G)|} to the vertices of G such that the sum of labels on neighbors of u is the same for all vertices u. It is known that the n-dimensional hypercube Q_n has a distance magic labeling if and only if n≡2(mod4). Let Γ be an Abelian group of order |V(G)|. Analogously, a Γ-distance magic labeling of G is a bijection ℓ:V→Γ for which the sum of labels on neighbors of u is the same for all vertices u. In this paper we fully characterize Γ-distance magic labellings of n-dimensional hypercubes Q_n. Namely we prove that for n odd, there does not exist a Γ-distance magic labeling of Q_n for any Abelian group Γ of order |V(Q_n)|, whereas for n even there exists a Γ-distance magic labeling of Q_n for every Abelian group Γ of order |V(Q_n)|. Similarly distance antimagic and Γ-distance antimagic labellings can be defined, where one aims to find a bijection such that the sums of labels are pairwise distinct for all the vertices. We study this problem and show in particular that there exists a Γ-distance antimagic labeling of Q_n for any Abelian group Γ of order 2ⁿ if n is odd. We also indicate some relationships between Γ-closed distance magic and antimagic labellings and Γ-distance antimagic labellings. The proofs rely mostly on linear algebra.