Group distance magic Cartesian product of two cycles

Sylwia Cichacz, Paweł Dyrlaga, Dalibor Froncek

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Article number111807
JournalDiscrete Mathematics
Volume343
Issue number5
DOIs
StatePublished - May 2020

Keywords

  • Abelian group
  • Cartesian product
  • Distance magic labeling

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