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 λ : V ∪E → {1, 2, . . . , |V | + |E|} so that for every x ∈ V , w(x) = λ(x)+ ∑ xy∈E λ(xy) = 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, . . . , |V |. In this paper we introduce a new method to expand some known VMT labelings of 2-regular graphs.