Group distance magic labeling of direct product of graphs
Marcin Anholcer, Sylwia Cichacz, Iztok Peterin, Aleksandra Tepeh · Ars Mathematica Contemporanea · 2014
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 C m × G is Γ -distance magic for every Abelian group of order m n . We also prove that C m × C n is Z m n -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 C m × C n is not Γ -distance magic for any Abelian group Γ of order m n .