Geodesic
On the uniqueness of $D$-vertex magic constant
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 279-286.

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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 N_D(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.
Keywords: distance magic graph, D-vertex magic graph, magic constant, dominating function, fractional domination number 
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Arumugam, S.; Kamatchi, N.; Vijayakumar, G.R. On the uniqueness of $D$-vertex magic constant. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 279-286. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a5/

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