In particular, we obtained a homeomorphism [M.sub.5]([pi]/2) [congruent to] [[summation].sub.5], where [[summation].sub.5] denotes a connected closed

orientable surface of genus 5.

For i = 0,1, 2, ..., let [g.sub.i](G) be the number of topologically distinct cellular embeddings of the graph G in the

orientable surface [S.sub.i] of genus i.

A map is a connected graph embedded in a compact connected

orientable surface in such a way that the regions delimited by the graph, called faces, are homeomorphic to open discs.

Let M be a 3-manifold which is constructed as a fiber bundle over a closed

orientable surface with fiber a circle.

Throughout this paper, S will be an

orientable surface from [R.sup.3.sub.1] and n its unit normal vector field.

Certainly, on an

orientable surface, an equivalence preserve the orientation on this surface.

Henceforth, by surface we shall always mean an

orientable surface, equivalently, one which is embeddable in [R.sup.3].

A map M is an embedding of a connected graph G, with possibly multi-edges or loops, into a closed, connected and

orientable surface S such that all faces, i.e.

This graph is embedded in a locally

orientable surface such that if we cut the graph from the surface, the remaining part consists of connected components called faces or cells, each homeomorphic to an open disk.

A unicellular map is a graph embedded on a compact

orientable surface, in such a way that its complement is a topological polygon.

Bender and Canfield ([BC86]) obtained the asymptotic number of maps on a given

orientable surface. Gao ([Gao93]) obtained formulas for the asymptotic number of 21-angulations on

orientable surfaces, and conjectured a formula for more general families (namely maps where the degrees of the faces are restricted to lie in a given finite subset of 2N).