Vertex magic total labelings of 2-regular graphs

A vertex magic total (VMT) labeling of a graph $G=(V,E)$ is a bijection from the set of vertices and edges to the set of integers defined by $\lambda :V\cup E\to \{1,2,\dots ,|V|+|E\left|\right\}$ so that for every $x\in V$, $w\left(x\right)=\lambda \left(x\right)+{\sum }_{xy\in E}\lambda \left(xy\right)=k$, for some integer $k$. A VMT labeling is said to be a super VMT labeling if the vertices are labeled with the smallest possible integers, $1,2,\dots ,\left|V\right|$. In this paper we introduce a new method to expand some known VMT labelings of 2-regular graphs.