The genus of complete 3-uniform hypergraphs

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 $V(H)\sqcup E(H)$,inwhich$v\in V(H)$and$e\in E(H)$areadjacentifandonlyif$v$and$e$areincidentin$H$. 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.