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 $\mathrm{1,2},\dots ,|E(G)|$ 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.