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Journal of Applied Mathematics
Volume 2015 (2015), Article ID 320616, 7 pages
http://dx.doi.org/10.1155/2015/320616
Research Article

On -Vertex-Antimagic Edge Labeling of Regular Graphs

1Department of Applied Mathematics and Informatics, Technical University, Letná 9, 04200 Košice, Slovakia
2Department of Applied Mathematics, Tunghai University, Taichung 40704, Taiwan
3Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan

Received 1 April 2015; Accepted 26 May 2015

Academic Editor: Heping Zhang

Copyright © 2015 Martin Bača et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.