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 ∑ x∈N(v) l(x) = k for all v ∈ V , where N(v) is the open neighborhood of v. In this paper we deal with circulant graphs C(1, p). The circulant graph Cn(1, p) is the graph on the vertex set V = {x0, x1, . . . , xn−1} with edges (xi, xi+p) for i = 0, . . . , n−1 where i + p is taken modulo n. We completely characterize distance magic graphs Cn(1, p) for p odd. We also give some sufficient conditions for p even. Moreover, we also consider a group distance magic labeling of Cn(1, p).