Distance magic graphs of high regularity

Abstract

A distance magic labeling of a graph G with n vertices is such a bijection f from the vertex set of G to the set of integers {1, 2, …, n} that for every vertex in G the sum of labels of all adjacent vertices gives the same value k. A graph that allows such a labeling is a distance magic graph.There is an elegant construction of r-regular distance magic graphs with an even number of vertices for all feasible values of r. For graphs of odd order certain necessary and certain sufficient conditions are known for the existence of a distance magic labeling. In this paper we show that an (n -3)-regular distance magic graph with n vertices exists iff n ≡ 3 (mod 6) and that its structure is determined uniquely.Moreover, we simplify the constructions from a recent paper by Fronček into a single construction and provide another sufficient condition for the existence a distance magic graph with an odd number of vertices.