Vertex magic total labelings of 2-regular graphs

A vertex magic total (VMT) labeling   of a graph G=(V,E)G=(V,E) is a bijection from the set of vertices and edges to the set of integers defined by λ:V∪E→{1,2,…,|V|+|E|}λ:V∪E→{1,2,…,|V|+|E|} so that for every x∈Vx∈V, w(x)=λ(x)+∑xy∈Eλ(xy)=kw(x)=λ(x)+∑xy∈Eλ(xy)=k, for some integer kk. A VMT labeling is said to be a super   VMT labeling if the vertices are labeled with the smallest possible integers, 1,2,…,|V|1,2,…,|V|. In this paper we introduce a new method to expand some known VMT labelings of 2-regular graphs.