Note on group distance magic complete bipartite graphs

Abstract

A Γ-distance magic labeling of a graph G = (V, E) with {pipe}V{pipe} = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight w(x) = ∑y∈NG(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 {pipe}V(G){pipe}. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4). © 2014 Versita Warsaw and Springer-Verlag Wien.