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 ) + ∑ _{x y ∈ E} λ ( x y ) = 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.