# Vertex-antimagic total labelings of graphs

• Volume: 23, Issue: 1, page 67-83
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Martin Bača, et al. "Vertex-antimagic total labelings of graphs." Discussiones Mathematicae Graph Theory 23.1 (2003): 67-83. <http://eudml.org/doc/270428>.

@article{MartinBača2003,
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. },
author = {Martin Bača, James A. MacDougall, François Bertault, Mirka Miller, Rinovia Simanjuntak, Slamin},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {super-magic labeling; (a,d)-vertex-antimagic total labeling; (a,d)-antimagic labeling; graph labeling; antimagic labeling; -vertex-antimagic total labeling},
language = {eng},
number = {1},
pages = {67-83},
title = {Vertex-antimagic total labelings of graphs},
url = {http://eudml.org/doc/270428},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Martin Bača
AU - James A. MacDougall
AU - François Bertault
AU - Mirka Miller
AU - Rinovia Simanjuntak
AU - Slamin
TI - Vertex-antimagic total labelings of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2003
VL - 23
IS - 1
SP - 67
EP - 83
AB - 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.
LA - eng
KW - super-magic labeling; (a,d)-vertex-antimagic total labeling; (a,d)-antimagic labeling; graph labeling; antimagic labeling; -vertex-antimagic total labeling
UR - http://eudml.org/doc/270428
ER -

## References

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10. [10] J.A. MacDougall, M. Miller, Slamin, and W.D. Wallis, Vertex-magic total labelings of graphs, Utilitas Math., to appear. Zbl1008.05135
11. [11] M. Miller and M. Bača, Antimagic valuations of generalized Petersen graphs, Australasian J. Combin. 22 (2000) 135-139. Zbl0971.05098
12. [12] J. Sedlácek, Problem 27 in Theory of Graphs and its Applications, Proc. Symp. Smolenice, June 1963, Praha (1964), p. 162.
13. [13] B.M. Stewart, Supermagic complete graphs, Can. J. Math. 19 (1967) 427-438, doi: 10.4153/CJM-1967-035-9. Zbl0162.27801
14. [14] W.D. Wallis, E.T. Baskoro, M. Miller and Slamin, Edge-magic total labelings of graphs, Australasian J. Combin. 22 (2000) 177-190. Zbl0972.05043
15. [15] D.B. West, An Introduction to Graph Theory (Prentice-Hall, 1996). Zbl0845.05001

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