102 research outputs found
Enumeration of labelled 4-regular planar graphs
We present the first combinatorial scheme for counting labelled 4-regular
planar graphs through a complete recursive decomposition. More precisely, we
show that the exponential generating function of labelled 4-regular planar
graphs can be computed effectively as the solution of a system of equations,
from which the coefficients can be extracted. As a byproduct, we also enumerate
labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps
Planar maps as labeled mobiles
We extend Schaeffer's bijection between rooted quadrangulations and
well-labeled trees to the general case of Eulerian planar maps with prescribed
face valences, to obtain a bijection with a new class of labeled trees, which
we call mobiles. Our bijection covers all the classes of maps previously
enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the
enumeration of maps to that, much simpler, of mobiles and moreover keeps track
of the geodesic distance within the initial maps via the mobiles' labels.
Generating functions for mobiles are shown to obey systems of algebraic
recursion relations.Comment: 31 pages, 17 figures, tex, lanlmac, epsf; improved tex
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
Simple recurrence formulas to count maps on orientable surfaces
We establish a simple recurrence formula for the number of rooted
orientable maps counted by edges and genus. We also give a weighted variant for
the generating polynomial where is a parameter taking the number
of faces of the map into account, or equivalently a simple recurrence formula
for the refined numbers that count maps by genus, vertices, and
faces. These formulas give by far the fastest known way of computing these
numbers, or the fixed-genus generating functions, especially for large . In
the very particular case of one-face maps, we recover the Harer-Zagier
recurrence formula.
Our main formula is a consequence of the KP equation for the generating
function of bipartite maps, coupled with a Tutte equation, and it was
apparently unnoticed before. It is similar in look to the one discovered by
Goulden and Jackson for triangulations, and indeed our method to go from the KP
equation to the recurrence formula can be seen as a combinatorial
simplification of Goulden and Jackson's approach (together with one additional
combinatorial trick). All these formulas have a very combinatorial flavour, but
finding a bijective interpretation is currently unsolved.Comment: Version 3: We changed the title once again. We also corrected some
misprints, gave another equivalent formulation of the main result in terms of
vertices and faces (Thm. 5), and added complements on bivariate generating
functions. Version 2: We extended the main result to include the ability to
track the number of faces. The title of the paper has been changed
accordingl
A bijection for triangulations, quadrangulations, pentagulations, etc
A -angulation is a planar map with faces of degree . We present for
each integer a bijection between the class of -angulations of
girth (i.e., with no cycle of length less than ) and a class of
decorated plane trees. Each of the bijections is obtained by specializing a
"master bijection" which extends an earlier construction of the first author.
Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for
triangulations () and by Schaeffer for quadrangulations (). For
, both the bijections and the enumerative results are new. We also
extend our bijections so as to enumerate \emph{-gonal -angulations}
(-angulations with a simple boundary of length ) of girth . We thereby
recover bijectively the results of Brown for simple -gonal triangulations
and simple -gonal quadrangulations and establish new results for .
A key ingredient in our proofs is a class of orientations characterizing
-angulations of girth . Earlier results by Schnyder and by De Fraysseix
and Ossona de Mendez showed that simple triangulations and simple
quadrangulations are characterized by the existence of orientations having
respectively indegree 3 and 2 at each inner vertex. We extend this
characterization by showing that a -angulation has girth if and only if
the graph obtained by duplicating each edge times admits an orientation
having indegree at each inner vertex
Uniform random sampling of planar graphs in linear time
This article introduces new algorithms for the uniform random generation of
labelled planar graphs. Its principles rely on Boltzmann samplers, as recently
developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the
Boltzmann framework, a suitable use of rejection, a new combinatorial bijection
found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic
description of the generating functions counting planar graphs, which was
recently obtained by Gim\'enez and Noy. This gives rise to an extremely
efficient algorithm for the random generation of planar graphs. There is a
preprocessing step of some fixed small cost. Then, the expected time complexity
of generation is quadratic for exact-size uniform sampling and linear for
approximate-size sampling. This greatly improves on the best previously known
time complexity for exact-size uniform sampling of planar graphs with
vertices, which was a little over .Comment: 55 page
Enumeration of labeled 4-regular planar graphs
© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps.Peer ReviewedPostprint (author's final draft
Some families of increasing planar maps
Stack-triangulations appear as natural objects when one wants to define some
increasing families of triangulations by successive additions of faces. We
investigate the asymptotic behavior of rooted stack-triangulations with
faces under two different distributions. We show that the uniform distribution
on this set of maps converges, for a topology of local convergence, to a
distribution on the set of infinite maps. In the other hand, we show that
rescaled by , they converge for the Gromov-Hausdorff topology on
metric spaces to the continuum random tree introduced by Aldous. Under a
distribution induced by a natural random construction, the distance between
random points rescaled by converge to 1 in probability.
We obtain similar asymptotic results for a family of increasing
quadrangulations
A unified bijective method for maps: application to two classes with boundaries
International audienceBased on a construction of the first author, we present a general bijection between certain decorated plane trees and certain orientations of planar maps with no counterclockwise circuit. Many natural classes of maps (e.g. Eulerian maps, simple triangulations,...) are in bijection with a subset of these orientations, and our construction restricts in a simple way on the subset. This gives a general bijective strategy for classes of maps. As a non-trivial application of our method we give the first bijective proofs for counting (rooted) simple triangulations and quadrangulations with a boundary of arbitrary size, recovering enumeration results found by Brown using Tutte's recursive method.En nous appuyant sur une construction du premier auteur, nous donnons une bijection générale entre certains arbres décorés et certaines orientations de cartes planaires sans cycle direct. De nombreuses classes de cartes (par exemple les eulériennes, les triangulations) sont en bijection avec un sous-ensemble de ces orientations, et notre construction se spécialise de maniÚre simple sur le sous-ensemble. Cela donne un cadre bijectif général pour traiter les familles de cartes. Comme application non-triviale de notre méthode nous donnons les premiÚres preuves bijectives pour l'énumération des triangulations et quadrangulations simples (enracinées) ayant un bord de taille arbitraire, et retrouvons ainsi des formules de comptage trouvées par Brown en utilisant la méthode récursive de Tutte
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