# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## VERTEX-ANTIMAGIC TOTAL LABELINGS OF GRAPHS

 Martin Bača Department of Applied Mathematics Technical University, 04200 Košice, Slovak Republic e-mail: Martin.Baca@tuke.sk James A. MacDougall Department of Mathematics University of Newcastle, NSW 2308, Australia e-mail: jmacd@math.newcastle.edu.au François Bertault Department of Computer Science and Software Engineering University of Newcastle, NSW 2308, Australia e-mail: francois@cs.newcastle.edu.au Mirka Miller, Rinovia Simanjuntak and Slamin Department of Computer Science and Software Engineering University of Newcastle, NSW 2308, Australia e-mail: mirka@cs.newcastle.edu.au, rino@cs.newcastle.edu.au, slamin@cs.newcastle.edu.au

## Abstract

In this paper we introduce a new type of graph labeling for a graph G(V,E) called an (a,d)-vertex-antimagic total labeling. In this labeling we assign to the vertices and edges the consecutive integers from 1 to |V|+|E| and calculate the sum of labels at each vertex, i.e., the vertex label added to the labels on its incident edges. These sums form an arithmetical progression with initial term a and common difference d.

We investigate basic properties of these labelings, show their relationships with several other previously studied graph labelings, and show how to construct labelings for certain families of graphs. We conclude with several open problems suitable for further research.

Keywords: super-magic labeling, (a,d)-vertex-antimagic total labeling, (a,d)-antimagic labeling.

2000 Mathematics Subject Classification: 05C78, 05C05, 05C38.

Open problem 1 For the paths Pn and the cycles Cn, determine if there is a vertex-antimagic total labeling for every feasible pair (a,d).

Open problem 2 Apart from duality, how can a vertex-antimagic total labeling for a graph be used to construct another vertex-antimagic total labeling for the same graph, preferably with different a and d?

Open problem 3 In Theorem 3, we found a way to construct VATL for a graph G from a vertex-magic total labeling of G. Are there other ways to do this?

Open problem 4 Find, if possible, some structural characteristics of a graph which make a vertex-antimagic total labeling impossible.

## References

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