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

Abstract

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 | V | = n and | E | = e . A vertex-magic total labeling (VMTL) is a bijection λ from V ∪ E to the consecutive integers 1 , 2 , … , n + e with the property that for every v ∈ V , λ ( v ) + ∑ w ∈ N ( v ) λ ( v , w ) = h , for some constant h. Such a labeling is super if λ ( 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.