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.
AS everyone knows, so-called the plane of any planar map is pointed to the plane of Euclidean geometry.
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.
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 .
In particular, several papers have been devoted in the past 20 years to the study of the Potts model on families of planar maps [1, 7, 14, 16, 18, 26].
n] be a random planar map chosen uniformly at random in the space [M.
The resulting planar map M is a 2p-angulation, which is rooted at the oriented edge between [partial derivative] and [v.
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 the bijection between simple quadrangulations and 3-connected planar maps, and using Euler's relation, the GF xwQ(xz, z, w) counts rooted 3-connected planar maps, where z marks edges (we have added an extra term w to correct the 'minus one' in the definition of Q).
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.