Since a planar map from any spherical map contains all regional planar maps, hence we need merely to prove the whole planar map under any possible circumstances.

AS everyone knows, so-called the plane of any planar map is pointed to the plane of Euclidean geometry.

For any

planar map M with order [greater than or equal to] 3 and vertex valency [greater than or equal to] 3, there is an angle factor [mu] such that (M, [mu]) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5), (L5) and (R5).

Keywords Parallel bundle; Planar map; Smarandache geometry; Map geometry; Classification.

We classify parallel bundles in planar map geometries along an orientation [??].

A planar map is a proper embedding of a connected graph (possibly with loops and multiple edges) in the oriented sphere, considered up to continuous deformation.

Beyond planar maps, which are now well understood, the attention has also focussed on two more general objects: maps on higher genus surfaces, and maps equipped with an additional structure.

Let M be a planar map and let V (M) denote the vertex set of M.

The resulting planar map M is a 2p-angulation, which is rooted at the oriented edge between [partial derivative] and [v.sub.0] = 0, corresponding to i = 0 in the previous construction.

We remark that an analogous result holds for planar maps (11), counted according to the number of edges.

In (6), the authors studied the generating function h(x, y, w) of rooted non-separable planar maps where x, y and w count, respectively the number of vertices minus one, the number of faces minus one, and the valency (number of edges) of the external face.

By analytic techniques, involving recursive decompositions and non trivial manipulations of power series, Tutte obtained beautiful and simple enumerative formulas for several families of

planar maps. His techniques were extended in the late eighties by several authors to more sophisticated families of maps or to the case of maps of higher genus.