Group Magicness of Complete $N$-partite Graphs

Abstract

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 $K_{k_1,k_2,\dots ,k_n}$ (where $K_i\ge 2$) is $A$-magic, for all $A$ where $|A|\ge 3$.