Group Magicness of Complete N-partite Graphs

Let \( A \) be a non-trivial abelian group. We call a graph \( G = (V,E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A \setminus \{0\} \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum f(u,v) \) where the sum is over all \( (u,v) \in E(G) \), is a constant map. In this paper, we show that \( K_{k_1,k_2,\ldots,k_n} \) (where \( K_{i} \geq 2 \)) is \( A \)-magic, for all \( A \) where \( |A| \geq 3 \).