On (a,1)-Vertex-Antimagic Edge Labeling of Regular Graphs

Abstract

An (a,s)-vertex-antimagic edge labeling (or an (a,s)-VAE labeling, for short) of G is a bijective mapping from the edge set E(G) of a graph G to the set of integers 1,2,…,|E(G)| with the property that the vertex-weights form an arithmetic sequence starting from a and having common difference s, where a and s 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 (a,s)-antimagic if it admits an (a,s)-VAE labeling. In this paper, we investigate the existence of (a,1)-VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept (a,s)-vertex-antimagic edge deficiency, as an extension of (a,s)-VAE labeling, for measuring how close a graph is away from being an (a,s)-antimagic graph. Furthermore, the (a,1)-VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.