Non-magic Hypergraphs

TL;DR

This paper explores the generalization of magic labelings from squares to hypergraphs, providing algorithms to determine such labelings over abelian groups and applying them to classify certain configurations.

Contribution

It introduces algorithms for identifying magic labelings in hypergraphs over abelian groups and integers, extending the concept beyond traditional magic squares.

Findings

Developed an algorithm for hypergraph magic labelings over abelian groups

Extended the algorithm to determine magic labelings over integers

Counted magic configurations for specific hypergraph sizes

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 $n=7, \dots, 14$.