On the Enumeration of Hypermaps Which are Self-Equivalent with Respect to Reversing the Colors of Vertices

Free subgroups of free products and combinatorial hypermaps

“…The partial picture in Figure 1 features a (5, 6)-hypermap. By analogy with fullerene polyhedra, we shall call (5,6)-hypermaps fullerene hypermaps, for short. However, in this case fullerene hypermaps do not have any specific restrictions (apart from those coming from the number of darts) on their topological genus, and thus on the number of pentagonal faces.…”

“…We shall refer to a (2, 3)-hypermap as a triangulation (of an oriented surface), thus allowing identification of two sides of the same triangle. Figure 2 shows a triangulation of a torus with α = (1, 4)(2, 5) (3,6), σ = (1, 6, 2, 4, 3, 5), ϕ = (1, 2, 3) (4,5,6).…”