Bicontactual Regular Hypermaps

Abstract

Bicontactual regular maps (regular maps with the property that each face meets only two others) were introduced and classified by Wilson in 1985. This property generalizes to hypermaps (cellular embeddings of hypergraphs in compact surfaces) giving rise to three types of bicontactuality, namely, the edge-twin, the vertex-twin, and the alternate (the first two of which are the dual of each other). In 2003 Wilson and Breda classified (up to an isomorphism and duality) the bicontactual regular nonorientable hypermaps, leaving the orientable case open. In this paper we classify the bicontactual regular oriented hypermaps. Introduction.A map is a cellular embedding of a graph in a compact surface. A hypermap is a generalization of a map obtained by allowing edges to connect more than two vertices; in other words, a hypermap is a "cellular embedding" of a hypergraph (or rather, the bipartite graph whose white vertices, black vertices, and edges are the hyperedges, hypervertices, and hypervertex-hyperedge incidence pairs, respectively, of the hypergraph) G in a compact and connected surface S. A hypermap is orientable if the underlying surface S has no boundary and is orientable; is nonorientable if S has no boundary and is nonorientable; and with boundary if S has boundary and the underlying bipartite graph G has no vertices lying on the boundary. In what follows is a brief introduction to hypermaps. For a further and deeper reading on maps we refer the reader to [1, 6, 13] and on hypermaps to [8,10,14]. The notation used in this paper follows that in [2,4,5].An oriented hypermap is an orientable hypermap with a fixed orientation; algebraically it is described by a triple Q = (D; R, L) consisting of a finite set of "abstract" darts D and two permutations R, L of D that generate a transitive group M on(Q) = R, L on D, called the monodromy group of Q; the orbits of R, L, and RL are the hypervertices, hyperedges, and hyperfaces, respectively. Maps are hypermaps satisfying L 2 = 1 1 and when a hypermap is not a map (L 2 = 1) it is called a proper hypermap. Since hypergraphs can be represented by bipartite (multi)graphs, hypermaps H will be viewed as bipartite maps M b (also called Walsh bipartite maps [15]) whose automorphism groups preserve the bipartition (two-set