On magic graphs

Abstract

A (p, q)-graph G = (V,E) is said to be magic if there exists a bijection f: V ? E ? {1, 2, 3,..., p + q} such that for all edges uv of G, f(u) + f(v) + f(uv) is a constant. The minimum of all constants say, m(G), where the minimum is taken over all such bijections of a magic graph G, is called the magic strength of G. In this paper we define the maximum of all constants say, M(G), analogous to m(G), and introduce strong magic, ideal magic, weak magic labelings, and prove that some known classes of graphs admit such labelings.