The genus of complete 3-uniform hypergraphs

Abstract

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph Kn. In this paper, we investigate the minimum genus embeddings of the complete 3-uniform hypergraphs Kn3. Embeddings of a hypergraph H are defined as the embeddings of its associated Levi graph LH with vertex set V(H)⊔E(H), in which v∈V(H) and e∈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 Kn3 when n is even. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of Kn3 is at least 214n2log⁡n(1−o(1)). The construction in the proof may be of independent interest as a design-type problem.