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.

The proof in the general case is essentially the same, we glue [[summation].sub.1], ..., [[summation].sub.k] cylinders to the surface [M.sub.1] to obtain an

orientable surface that is homeomorphic to M.

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).