1,444 research outputs found
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
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
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
On the two-point function of general planar maps and hypermaps
We consider the problem of computing the distance-dependent two-point
function of general planar maps and hypermaps, i.e. the problem of counting
such maps with two marked points at a prescribed distance. The maps considered
here may have faces of arbitrarily large degree, which requires new bijections
to be tackled. We obtain exact expressions for the following cases: general and
bipartite maps counted by their number of edges, 3-hypermaps and
3-constellations counted by their number of dark faces, and finally general and
bipartite maps counted by both their number of edges and their number of faces.Comment: 32 pages, 17 figure
Unified bijections for maps with prescribed degrees and girth
This article presents unified bijective constructions for planar maps, with
control on the face degrees and on the girth. Recall that the girth is the
length of the smallest cycle, so that maps of girth at least are
respectively the general, loopless, and simple maps. For each positive integer
, we obtain a bijection for the class of plane maps (maps with one
distinguished root-face) of girth having a root-face of degree . We then
obtain more general bijective constructions for annular maps (maps with two
distinguished root-faces) of girth at least . Our bijections associate to
each map a decorated plane tree, and non-root faces of degree of the map
correspond to vertices of degree of the tree. As special cases we recover
several known bijections for bipartite maps, loopless triangulations, simple
triangulations, simple quadrangulations, etc. Our work unifies and greatly
extends these bijective constructions. In terms of counting, we obtain for each
integer an expression for the generating function
of plane maps of girth with root-face of
degree , where the variable counts the non-root faces of degree .
The expression for was already obtained bijectively by Bouttier, Di
Francesco and Guitter, but for the expression of is new. We
also obtain an expression for the generating function
\G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees
and , such that cycles separating the two root-faces have length at
least while other cycles have length at least . Our strategy is to
obtain all the bijections as specializations of a single "master bijection"
introduced by the authors in a previous article. In order to use this approach,
we exhibit certain "canonical orientations" characterizing maps with prescribed
girth constraints
A simple formula for the series of constellations and quasi-constellations with boundaries
We obtain a very simple formula for the generating function of bipartite
(resp. quasi-bipartite) planar maps with boundaries (holes) of prescribed
lengths, which generalizes certain expressions obtained by Eynard in a book to
appear. The formula is derived from a bijection due to Bouttier, Di Francesco
and Guitter combined with a process (reminiscent of a construction of Pitman)
of aggregating connected components of a forest into a single tree. The formula
naturally extends to -constellations and quasi--constellations with
boundaries (the case corresponding to bipartite maps).Comment: 23 pages, full paper version of v1, with results extended to
constellations and quasi constellation
The enumeration of generalized Tamari intervals
Let be a grid path made of north and east steps. The lattice
, based on all grid paths weakly above and
sharing the same endpoints as , was introduced by Pr\'eville-Ratelle and
Viennot (2014) and corresponds to the usual Tamari lattice in the case
. Our main contribution is that the enumeration of intervals in
, over all of length , is given by . This formula was first obtained by Tutte(1963) for
the enumeration of non-separable planar maps. Moreover, we give an explicit
bijection from these intervals in to non-separable
planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From
generalized Tamari intervals to non-separable planar maps (extended
abstract)", submitte
The Bernardi process and torsor structures on spanning trees
Let G be a ribbon graph, i.e., a connected finite graph G together with a
cyclic ordering of the edges around each vertex. By adapting a construction due
to O. Bernardi, we associate to any pair (v,e) consisting of a vertex v and an
edge e adjacent to v a bijection between spanning trees of G and elements of
the set Pic^g(G) of degree g divisor classes on G, where g is the genus of G.
Using the natural action of the Picard group Pic^0(G) on Pic^g(G), we show that
the Bernardi bijection gives rise to a simply transitive action \beta_v of
Pic^0(G) on the set of spanning trees which does not depend on the choice of e.
A plane graph has a natural ribbon structure (coming from the
counterclockwise orientation of the plane), and in this case we show that
\beta_v is independent of v as well. Thus for plane graphs, the set of spanning
trees is naturally a torsor for the Picard group. Conversely, we show that if
\beta_v is independent of v then G together with its ribbon structure is
planar. We also show that the natural action of Pic^0(G) on spanning trees of a
plane graph is compatible with planar duality.
These findings are formally quite similar to results of Holroyd et al. and
Chan-Church-Grochow, who used rotor-routing to construct an action r_v of
Pic^0(G) on the spanning trees of a ribbon graph G, which they show is
independent of v if and only if G is planar. It is therefore natural to ask how
the two constructions are related. We prove that \beta_v = r_v for all vertices
v of G when G is a planar ribbon graph, i.e. the two torsor structures
(Bernardi and rotor-routing) on the set of spanning trees coincide. In
particular, it follows that the rotor-routing torsor is compatible with planar
duality. We conjecture that for every non-planar ribbon graph G, there exists a
vertex v with \beta_v \neq r_v.Comment: 25 pages. v2: numerous revisions based on referee comments. v3:
substantial additional revisions; final version to appear in IMR
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