Geodesic
Group distance magic labeling of direct product of graphs
Ars Mathematica Contemporanea, Tome 9 (2015) no. 1, pp. 93-107.

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Let G = (V, E) be a graph and Γ  an Abelian group, both of order n. A group distance magic labeling of G is a bijection ℓ: V → Γ  for which there exists μ ∈ Γ  such that ∑ x ∈ N(v)ℓ(x) = μ for all v ∈ V, where N(v) is the neighborhood of v. In this paper we consider group distance magic labelings of direct product of graphs. We show that if G is an r -regular graph of order n and m = 4 or m = 8 and r is even, then the direct product Cm × G is Γ -distance magic for every Abelian group of order mn. We also prove that Cm × Cn is Zmn-distance magic if and only if m ∈ {4, 8} or n ∈ {4, 8} or m, n ≡ 0 mod 4. It is also shown that if m, n not≡ 0 mod 4 then Cm × Cn is not Γ -distance magic for any Abelian group Γ  of order mn.
DOI : 10.26493/1855-3974.432.2c9
Keywords: Distance magic labeling, group labeling, strong product of graphs 
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Marcin Anholcer; Cichacz Sylwia; Iztok Peterin; Aleksandra Tepeh. Group distance magic labeling of direct product of graphs. Ars Mathematica Contemporanea, Tome 9 (2015) no. 1, pp. 93-107. doi : 10.26493/1855-3974.432.2c9. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.432.2c9/

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