We present the first polynomial uniform random sampling algorithm for simple branched coverings of degree n of the sphere by itself. More precisely, our algorithm generates in linear time increasing quadrangulations, which are equivalent combinatorial structures. Our result is based on the identification of some canonical labelled spanning trees, and yields a constructive proof of a celebrated formula of Hurwitz for the number of some factorizations of permutations in transpositions. The previous approaches were either non constructive or lead to exponential time algorithms for the sampling problem.

The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the 1960s. Following the bijective approach initiated by Cori and Vauquelin in the '80s, we describe a bijection between rooted maps, or rooted bipartite quadrangulations, on a surface of genus g and some simpler objects that generalize plane trees. Thanks to a rerooting argument, our bijection allows us to compute the generating series of rooted maps on a surface of genus g with respect to the number of edges, and to recover the asymptotic numbers of such maps. Our construction allows us to keep track in a bipartite quadrangulation of the distances of all vertices to a random basepoint. This is an analogue for higher genus surfaces of the basic result on which were built the recent advances in the comprehension of the intrinsic geometry of large random planar maps, hopefully opening the way to the study of a model of continuum random surfaces of genus g.

We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees with interesting consequences for enumeration, mesh compression, and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best-known uniform random sampler for labelled planar graphs. It also provides an encoding for the set P ( n ) of n -edge 3-connected planar graphs that matches the entropy bound 1/ n log 2 | P ( n )| = 2 + o (1) bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the optimal parametric rate 1/ n log 2 | P ( n , i , j )| bpe for graphs of P ( n ) with i vertices and j faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated with the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is, for instance, a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs.

This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations.

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