The genus of complete 3-uniform hypergraphs

TL;DR

This paper determines the minimum genus embeddings of complete 3-uniform hypergraphs, providing exact values for even n and showing an exponential lower bound on the number of such embeddings, advancing topological graph theory.

Contribution

It extends the understanding of hypergraph embeddings by calculating genus values for complete 3-uniform hypergraphs and establishing a lower bound on their embedding diversity.

Findings

Exact orientable and non-orientable genus for even n

At least exponential number of non-isomorphic minimum genus embeddings

Construction method of embeddings of independent interest

Abstract

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph $K_n$. In this paper, we investigate the minimum genus embeddings of the complete $3$-uniform hypergraphs $K_n^3$. Embeddings of a hypergraph $H$ are defined as the embeddings of its associated Levi graph $L_H$ with vertex set $V(H)\sqcup E(H)$, in which $v\in V(H)$ and $e\in E(H)$ are adjacent if and only if $v$ and $e$ are incident in $H$. We determine both the orientable and the non-orientable genus of $K_n^3$ when $n$ is even. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of $K_n^3$ is at least $2^{\frac{1}{4}n^2\log n(1-o(1))}$. The construction in the proof may be of independent interest as a design-type problem.