## Journal of Applied Mathematics

### 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 $1,2,\dots ,|E(G)|$ 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.

#### Article information

Source
J. Appl. Math., Volume 2015 (2015), Article ID 320616, 7 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.jam/1439816406

Digital Object Identifier
doi:10.1155/2015/320616

Mathematical Reviews number (MathSciNet)
MR3358357

#### Citation

Bača, Martin; Semaničová-Feňovčíková, Andrea; Wang, Tao-Ming; Zhang, Guang-Hui. On $(a,1)$ -Vertex-Antimagic Edge Labeling of Regular Graphs. J. Appl. Math. 2015 (2015), Article ID 320616, 7 pages. doi:10.1155/2015/320616. https://projecteuclid.org/euclid.jam/1439816406

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