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 eΓ such that £∗eiv(.,) ℓ(x) = μ for all v eV, where N(v) is the neighborhood of v. Froncek showed that the Cartesian product CmUCn, 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 CmUCnhas Z« x A-magic labeling for any α = 0 (mod lcm(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 this problem by proving some necessary conditions. We further prove that for n even the graph CnUCnhas 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 C2DC2. We also give necessary and sufficient condition for CmUCnwith gcd(m,n) = 1 to be Γ-distance magic.
|Original language||English (US)|
|State||Published - May 13 2019|