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.
In addition, any planar map must be filled from figures, yet can't have any empty spacing without figures.
We investigate the behaviors of parallel bundles in the planar map geometries.
A general condition for a family of parallel lines passing through a cut of a planar map being a parallel bundle is the following.
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.
For instance, one has studied maps equipped with a polymer, with an Ising model, with a proper coloring, with a loop model, with a spanning tree, percolation on planar maps .
Let M be a planar map and let V (M) denote the vertex set of M.
n] be a random planar map chosen uniformly at random in the space [M.
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.
A few years later, Schaeffer ([S699]), following the work of Cori and Vauquelin ([CV81]), gave in his thesis a bijection between planar maps and certain labeled trees which enables to recover the formulas of Tutte, and explains combinatorially their remarquable simplicity.