Bipartite-Uniform Hypermaps on the Sphere

Abstract

A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that "neighbouring" hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, all the elements have the same valency. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hypermap is regular. In this paper we classify the bipartite-uniform hypermaps on the sphere (up to duality). Two constructions of bipartite-uniform hypermaps are given. All bipartite-uniform spherical hypermaps are shown to be constructed in this way. As a by-product we show that every bipartite-uniform hypermap ${\cal H}$ on the sphere is bipartite-regular. We also compute their irregularity group and index, and also their closure cover ${\cal H}^{\Delta}$ and covering core ${\cal H}_{\Delta}$.