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

Let G=(V,E)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|V|=n and |E|=e|E|=e. A vertex-magic total labeling (VMTL) is a bijection λλ from V∪EV∪E to the consecutive integers 1,2,…,n+e1,2,…,n+e with the property that for every v∈Vv∈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}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 G0G0 which admits a super VMTL; for instance, the graph kG  , provides many super VMTLs for the graphs obtained from G0G0 by means of the addition to it of various sets of edges.