Distance magic labeling in complete 4-partite graphs

الأصل · EN

Let G be a complete k-partite simple undirected graph with parts of sizes p₁≤ p₂...≤ pₖ. Let Pⱼ=∑ᵢ₌₁ʲpᵢ for j=1,...,k. It is conjectured that G has distance magic labeling if and only if ∑ᵢ₌₁ᵖʲ (n-i+1)≥ jn+12/k for all j=1,...,k. The conjecture is proved for k=4, extending earlier results for k=2,3.