441 research outputs found
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
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 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
Schnyder decompositions for regular plane graphs and application to drawing
Schnyder woods are decompositions of simple triangulations into three
edge-disjoint spanning trees crossing each other in a specific way. In this
article, we define a generalization of Schnyder woods to -angulations (plane
graphs with faces of degree ) for all . A \emph{Schnyder
decomposition} is a set of spanning forests crossing each other in a
specific way, and such that each internal edge is part of exactly of the
spanning forests. We show that a Schnyder decomposition exists if and only if
the girth of the -angulation is . As in the case of Schnyder woods
(), there are alternative formulations in terms of orientations
("fractional" orientations when ) and in terms of corner-labellings.
Moreover, the set of Schnyder decompositions on a fixed -angulation of girth
is a distributive lattice. We also show that the structures dual to
Schnyder decompositions (on -regular plane graphs of mincut rooted at a
vertex ) are decompositions into spanning trees rooted at such
that each edge not incident to is used in opposite directions by two
trees. Additionally, for even values of , we show that a subclass of
Schnyder decompositions, which are called even, enjoy additional properties
that yield a reduced formulation; in the case d=4, these correspond to
well-studied structures on simple quadrangulations (2-orientations and
partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder
decompositions yields (planar) orthogonal and straight-line drawing algorithms.
For a 4-regular plane graph of mincut 4 with vertices plus a marked
vertex , the vertices of are placed on a grid according to a permutation pattern, and in the orthogonal drawing
each of the edges of has exactly one bend. Embedding
also the marked vertex is doable at the cost of two additional rows and
columns and 8 additional bends for the 4 edges incident to . We propose a
further compaction step for the drawing algorithm and show that the obtained
grid-size is strongly concentrated around for a uniformly
random instance with vertices
‘A Global History of Humanity’ : a high school textbook to change the world
This paper aims at presenting the forthcoming school textbook ‘A Global History of Humanity’ that spans from 70.000 BCE till the 21st century and narrates a global history of our world assuming a non-Eurocentric and non-nationalist perspective. The textbook covers the history of humanity through three volumes, combining a chronological and a thematic approach. Each volume is divided into three chronological chapters. Each chapter presents the four themes in which the textbook is structured: humans change nature; humans on the move; social organization and inequality; worldviews. The last part of this paper ties the long history of humanity narrated through the textbook to today’s central questions, discussing the conditions in which we find ourselves today and the challenges we are facing in the coming years
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 manière 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 premières 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
On the distance-profile of random rooted plane graphs
International audienceWe study the distance-profile of the random rooted plane graph Gn with n edges (by a plane graph we mean a planar map with no loops nor multiple edges). Our main result is that the profile and radius of Gn (with respect to the root-vertex), rescaled by (2n) 1/4 , converge to explicit distributions related to the Brownian snake. A crucial ingredient of our proof is a bijection we have recently introduced between rooted outer-triangular plane graphs and rooted eulerian triangulations, combined with ingredients from Chassaing and Schaeffer (2004), Bousquet-Mélou and Schaeffer (2000), and Addario-Berry and Albenque (2013). We also show that the result for plane graphs implies similar results for random rooted loopless maps and general maps
A bijection for plane graphs and its applications
International audienceThis paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations. 1 Introduction A planar graph is a graph that can be embedded in the plane (drawn in the plane without edge crossing). A pla-nar map is an embedding of a connected planar graph considered up to deformation. The enumeration of pla-nar maps has been the subject of intense study since the seminal work of Tutte in the 60's [20] showing that many families of planar maps have beautiful counting formulas. Starting with the work of Cori and Vauquelin [10] and then Schaeffer [18, 19], bijective constructions have been discovered that provide more transparent proofs of such formulas. The enumeration of planar graphs has also been the focus of a lot of efforts, culminating with the asymptotic counting formulas obtained by Giménez and Noy [16]. In this paper we focus on simple planar maps (planar maps without loops nor multiple edges), which are also called plane graphs. This family of planar maps has, quite surprisingly, not been considered until fairly recently. This is probably due to the fact that loops and multiple edges are typically allowed in studies about planar maps, whereas they are usually forbidden in studies about planar graphs. At any rate, the first result about plane graphs was an exact algebraic expressio
Regulation of the Permeability Transition Pore in Skeletal Muscle Mitochondria MODULATION BY ELECTRON FLOW THROUGH THE RESPIRATORY CHAIN COMPLEX I
We have investigated the regulation of the permeability transition pore (PTP), a cyclosporin A-sensitive channel, in rat skeletal muscle mitochondria. As is the case with mitochondria isolated from a variety of sources, skeletal muscle mitochondria can undergo a permeability transition following Ca2+uptake in the presence of Pi. We find that the PTP opening is dramatically affected by the substrates used for energization, in that much lower Ca2+ loads are required when electrons are provided to complex I rather than to complex II or IV. This increased sensitivity of PTP opening does not depend on differences in membrane potential, matrix pH, Ca2+ uptake, oxidation-reduction status of pyridine nucleotides, or production of H2O2, but is directly related to the rate of electron flow through complex I. Indeed, and with complex I substrates only, pore opening can be observed when depolarization is induced with uncoupler (increased electron flow) but not with cyanide (decreased electron flow). Consistent with pore regulation by electron flow, we find that PTP opening is inhibited by ubiquinone 0 at concentrations that partially inhibit respiration and do not depolarize the inner membrane. These data allow identification of a novel site of regulation of the PTP, suggest that complex I may be part of the pore complex, and open new perspectives for its pharmacological modulation in living cells
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