Constructions of bipartite and bipartite-regular hypermaps

TL;DR

This paper explores the properties of bipartite hypermaps, generalizing previous algebraic constructions, and demonstrates that all surfaces admit bipartite-regular hypermaps.

Contribution

It extends algebraic constructions of bipartite hypermaps and proves the existence of bipartite-regular hypermaps on all surfaces.

Findings

Properties of algebraic constructions of bipartite hypermaps

Generalization of Walsh representation for hypermaps

Existence of bipartite-regular hypermaps on all surfaces

Abstract

A hypermap is bipartite if its set of flags can be divided into two parts A and B so that both A and B are the union of vertices, and consecutive vertices around an edge or a face are contained in alternate parts. A bipartite hypermap is bipartite-regular if its set of automorphisms is transitive on A and on B. In this paper we see some properties of the constructions of bipartite hypermaps described algebraically by Breda and Duarte which generalize the construction induced by the Walsh representation of hypermaps. As an application we show that all surfaces have bipartite-regular hypermaps.