UNIQUENESS OF VERTEX MAGIC CONSTANTS.

A graph G of order n is said to be ∑' labeled if there exist a bijection f : V (G) → [n] and a constant c such that for all v ∈ V(G), ∑u∈N[v]f(u) = c. The uniqueness of the constant c has been an open question that was posed by Prof. Arumugam at the 2010 IWOGL Conference in Duluth. The question of the uniqueness of such a constant can be extended to arbitrary neighborhoods and to arbitrary sets of labels. For a set D ⊂ N, let ND(v) = {u ∈ V(G): d(u, v) ∈ D}, and let W be a multiset of real numbers. Graph G is said to be (D, W)-vertex magic if there exists a bijection g : V(G) → W such that for all v ∈ V(G), ∑u∈ND(v)g(u) is a constant, called the (D, W)-vertex magic constant. In this paper we prove that, even for these more general conditions, a (D, W)-vertex magic constant will be unique. Moreover, the constant can be determined using a generalization of the fractional domination number of the graph. [ABSTRACT FROM AUTHOR]