Distance magic circulant graphs

Abstract

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 neighborhood of v. In this paper we deal with circulant graphs Cn(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).