A note on 4-regular distance magic graphs

Let G = (V,E) be a graph on n vertices. A bijection f: V → {1, 2,..., n} is called a distance magic labeling of G if there exists an integer k such that ∑ _u∈N(υ) f(u) = k for all υ ∈ V, where N(υ) is the set of all vertices adjacent to υ. The constant k is the magic constant of f and any graph which admits a distance magic labeling is a distance magic graph. In this paper we solve some of the problems posted in a recent survey paper on distance magic graph labelings by Arumugam et al. We classify all orders n for which a 4-regular distance magic graph exists and by this we also show that there exists a distance magic graph with k = 2t for every integer t ≥ 6.