# On the uniqueness of d-vertex magic constant

S. Arumugam; N. Kamatchi; G.R. Vijayakumar

Discussiones Mathematicae Graph Theory (2014)

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

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topS. 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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- [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] 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
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