Constructions of bipartite and bipartite-regular hypermaps (Q1953304)

Summary: 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 \textit{A. Breda d'Azevedo} and \textit{R. Duarte} [Electron. J. Comb. 14, No. 1, Research paper R5, 20 p. (2007; Zbl 1115.05021)] which generalize the construction induced by the Walsh representation of hypermaps. As an application we show that all surfaces have bipartite-regular hypermaps.