Distance magic circulant graphs

Let G=(V,E) be a graph of order n. A distance magic labeling of G is a bijection ℓ:V → {1,2,...,n} for which there exists a positive integer k such that Σ_x∈N(v) ℓ(x)=k for all v ∈ V, where N(v) is the neighborhood of v. In this paper we deal with circulant graphs C_n(1,p). The circulant graph C_n(1,p) is the graph on the vertex set V={x₀,x₁,...,x_n-1} with edges (x_i,x_i+p) for i=0,...,n-1 where i+p is taken modulo n. We completely characterize distance magic graphs C_n(1,p) for p odd. We also give some sufficient conditions for p even. Moreover, we also consider a group distance magic labeling of C_n(1,p).