Note on group distance magic complete bipartite graphs

Abstract

A Γ-distance magic labeling of a graph G = (V E) with |V | = is a bijection from V to an Abelian group Γ of order such that the weight ( ) = ∈N G ( ) ( ) of every vertex ∈ 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)|.In this paper we give necessary and sufficient conditions for complete -partite graphs of odd order to be Z -distance magic. Moreover we show that if ≡ 2 (mod 4) and is even, then there does not exist a group Γ of order such that there exists a Γ-distance labeling for a -partite complete graph of order . We also prove that K is a group distance magic graph if and only if + ≡ 2 (mod 4). MSC:05C25, 05C78

Distance Magic Labeling and Two Products of Graphs

“…Some graphs which are distance magic among (some) products can be seen in [2,3,9,10]. The following product cycle and product related results were proved by Miller, Rodger, and Simanjuntak.…”

On zero sum-partition of Abelian groups into three sets and group distance magic labeling

“…Therefore we can see that G is Γ-distance magic if and only if Γ has the CSP(t)-property. The following theorems were proven in [3].…”

Zero Sum Partition of Abelian Groups into Sets of the Same Order and its Applications

“…Notice that the constant sum partitions of a group Γ lead to complete multipartite Γ-distance magic labeled graphs. For instance, the partition {0}, {1, 2, 4}, {3, 5, 6} of the group Z7 with constant sum 0 leads to a Z7 -distance magic labeling of the complete tripartite graph K 1,3,3 (see [3,5]).…”

“…However a Zn -distance magic graph on n vertices is not necessarily a distance magic graph. Moreover, there are some graphs that are not distance magic while at the same time they are group distance magic (see [3]).…”