Let G=(V,E) be a graph and Γ an Abelian group both of order n. A Γ-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. Froncek showed that the Cartesian product Cm□Cn, m,n≥3 is a Zmn-distance magic graph if and only if mn is even. It is also known that if mn is even then Cm□Cn has Zα×A-magic labeling for any α≡0(modlcm(m,n)) and any Abelian group A of order mn∕α. However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution of this problem by proving some necessary conditions. We further prove that for n even the graph Cn□Cn has a Γ-distance magic labeling for any Abelian group Γ of order n2. Moreover we show that if m≠n, then there does not exist a (Z2)m+n-distance magic labeling of the Cartesian product C2m □C2n . We also give a necessary and sufficient condition for Cm□Cn with gcd(m,n)=1 to be Γ-distance magic.
Bibliographical noteFunding Information:
This work was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks within subsidy of Ministry of Science and Higher Education, Poland.
© 2020 The Authors
- Abelian group
- Cartesian product
- Distance magic labeling