A Γ-magic Rectangle Set and Group Distance Magic Labeling

摘要

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection l from V to an Abelian group Γ of order n such that the weight w(x) = ∑_(y∈N_G(x)) l(y) of every vertex x ∈ V is equal to the same element μ ∈ Γ called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. A Γ-magic rectangle set MRS_Γ(a, b; c) of order abc is a collection of c arrays (a × b) whose entries are elements of group Γ, each appearing once, with all row sums in every rectangle equal to a constant ω ∈ Γ and all column sums in every rectangle equal to a constant δ ∈ Γ. In the paper we show that if a and b are both even then MRS_Γ (a, b; c) exists for any Abelian group Γ of order abc. Furthermore we use this result to construct group distance magic labeling for some families of graphs.