Distance Magic Labeling in Complete 4-partite Graphs

Abstract

Let G be a complete k-partite simple undirected graph with parts of sizes $$p_1\le p_2\cdots \le p_k$$p1≤p2?≤pk. Let $$P_j=\sum _{i=1}^jp_i$$Pj=?i=1jpi for $$j=1,\ldots ,k$$j=1,?,k. It is conjectured that G has distance magic labeling if and only if $$\sum _{i=1}^{P_j} (n-i+1)\ge j{{n+1}\atopwithdelims (){2}}/k$$?i=1Pj(n-i+1)?jn+12/k for all $$j=1,\ldots ,k$$j=1,?,k. The conjecture is proved for $$k=4$$k=4, extending earlier results for $$k=2,3$$k=2,3.