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.
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A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ : V → {1, . . . , n} such that the weight of every vertex v, computed as the sum of the labels on the vertices in the open neighborhood of v, is a constant. In this paper, we show that hypercubes with dimension divisible by four are not distance magic. We also provide some positive results by proving necessary and sufficient conditions for the Cartesian product of certain complete multipartite graphs and the cycle on four vertices to be distance magic.
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A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ from V to the set {1, . . . , n} such that the weight w(x) = ∑y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.
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