Two new methods to obtain super vertex-magic total labelings of graphs

Let G = (V, E) be a finite non-empty graph, where V and E are the sets of vertices and edges of G, respectively, and vertical bar V vertical bar = n and vertical bar E vertical bar = e. A vertex-magic total labeling (VMTL) is a bijection lambda from V U E to the consecutive integers 1, 2, ..., n + e with the property that for every v is an element of V, lambda(v) + Sigma(w is an element of N(v)) lambda(v, w) = h, for some constant h. Such a labeling is super if lambda(V) = {1, 2, ..., n}. In this paper, two new methods to obtain super VMTLs of graphs are put forward. The first, from a graph G with some characteristics, provides a super VMTL to the graph kG graph composed by the disjoint unions of k copies of G, for a large number of values of k. The second, from a graph G(0) which admits a super VMTL; for instance, the graph kG, provides many super VMTLs for the graphs obtained from G(0) by means of the addition to it of various sets of edges. (C) 2007 Elsevier B.V. All rights reserved.