On -Vertex-Antimagic Edge Labeling of Regular Graphs

An - vertex-antimagic edge labeling (or an - VAE labeling, for short) of is a bijective mapping from the edge set of a graph to the set of integers with the property that the vertex-weights form an arithmetic sequence starting from and having common difference , where and are two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called -antimagic if it admits an - VAE labeling. In this paper, we investigate the existence of -VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept - vertex-antimagic edge deficiency , as an extension of -VAE labeling, for measuring how close a graph is away from being an -antimagic graph. Furthermore, the -VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.