ON THE UNIQUENESS OF D-VERTEX MAGIC CONSTANT

Let G = (V, E) be a graph of order n and let D subset of {0, I, 2, 3,...}. For v is an element of V, let N-D(v) = {u is an element of V : d(u, v) is an element of D}. The graph G is said to be D-vertex magic if there exists a bijection f : V(G) -> {I, 2,..., n} such that for all v is an element of v, Sigma(u is an element of ND(v)) f(u) is a constant, called D-vertex magic constant. O'Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r >= 4.