### Group distance magic labeling of direct product of graphs

#### Abstract

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_{mn}-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*.

#### Keywords

DOI: https://doi.org/10.26493/1855-3974.432.2c9

ISSN: 1855-3974

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