# On the uniqueness of d-vertex magic constant

• Volume: 34, Issue: 2, page 279-286
• ISSN: 2083-5892

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## Abstract

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Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.

## How to cite

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S. Arumugam, N. Kamatchi, and G.R. Vijayakumar. "On the uniqueness of d-vertex magic constant." Discussiones Mathematicae Graph Theory 34.2 (2014): 279-286. <http://eudml.org/doc/267976>.

@article{S2014,
abstract = {Let G = (V,E) be a graph of order n and let D ⊆ \{0, 1, 2, 3, . . .\}. For v ∈ V, let ND(v) = \{u ∈ V : d(u, v) ∈ D\}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → \{1, 2, . . . , n\} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.},
author = {S. Arumugam, N. Kamatchi, G.R. Vijayakumar},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {distance magic graph; D-vertex magic graph; magic constant; dominating function; fractional domination numbe; -vertex magic graph},
language = {eng},
number = {2},
pages = {279-286},
title = {On the uniqueness of d-vertex magic constant},
url = {http://eudml.org/doc/267976},
volume = {34},
year = {2014},
}

TY - JOUR
AU - S. Arumugam
AU - N. Kamatchi
AU - G.R. Vijayakumar
TI - On the uniqueness of d-vertex magic constant
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 2
SP - 279
EP - 286
AB - Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.
LA - eng
KW - distance magic graph; D-vertex magic graph; magic constant; dominating function; fractional domination numbe; -vertex magic graph
UR - http://eudml.org/doc/267976
ER -

## References

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1. [1] S. Arumugam, D. Fronček and N. Kamatchi, Distance magic graphs-A survey, J. Indones. Math. Soc., Special Edition (2011) 11-26. Zbl1288.05216
2. [2] S. Beena, On ∑ and ∑′ labelled graphs, Discrete Math. 309 (2009) 1783-1787. doi:10.1016/j.disc.2008.02.038[Crossref]
3. [3] G. Chartrand and L. Lesniak, Graphs & Digraphs, 4th Edition (Chapman and Hall, CRC, 2005).
4. [4] D. Grinstead and P.J. Slater, Fractional domination and fractional packings in graphs, Congr. Numer. 71 (1990) 153-172. Zbl0691.05043
5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., 1998). Zbl0883.00011
6. [6] M. Miller, C. Rodger and R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin. 28 (2003) 305-315. Zbl1031.05117
7. [7] A. O’Neal and P.J. Slater, An introduction to distance D magic graphs, J. Indones. Math. Soc., Special Edition (2011) 91-107. Zbl1288.05227
8. [8] A. O’Neal and P.J. Slater, Uniqueness of vertex magic constants, SIAM J. Discrete Math. 27 (2013) 708-716. doi:10.1137/110834421[Crossref][WoS] Zbl1272.05174
9. [9] K.A. Sugeng, D. Fronček, M. Miller, J. Ryan and J. Walker, On distance magic labeling of graphs, J. Combin. Math. Combin. Comput. 71 (2009) 39-48. Zbl1197.05133
10. [10] V. Vilfred, ∑-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, Trivandrum, India, 1994.

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