Distance magic labeling in complete 4-partite graphs

Abstract

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j} (n-i+1)\ge j{{n+1}\choose{2}}/k$ for all $j=1,...,k$. The conjecture is proved for $k=4$, extending earlier results for $k=2,3$.