Non-magic Hypergraphs

Abstract

This article studies a generalization of magic squares to $k$-uniform hypergraphs. In traditional magic squares the entries come from the natural numbers. A magic labeling of the vertices in a graph or hypergraph has since been generalized to allow for labels coming from any abelian group. We demonstrate an algorithm for determining whether a given hypergraph has a magic labeling over some abelian group. A slight adjustment of this algorithm also allows one to determine whether a given hypergraph can be magically labeled over $\mathbb{Z}$. As a demonstration, we use these algorithms to determine the number of magic $n_3$-configurations for $n=7, \dots, 14$.