479 research outputs found
A bijection for rooted maps on general surfaces
We extend the Marcus-Schaeffer bijection between orientable rooted bipartite
quadrangulations (equivalently: rooted maps) and orientable labeled one-face
maps to the case of all surfaces, that is orientable and non-orientable as
well. This general construction requires new ideas and is more delicate than
the special orientable case, but it carries the same information. In
particular, it leads to a uniform combinatorial interpretation of the counting
exponent for both orientable and non-orientable rooted
connected maps of Euler characteristic , and of the algebraicity of their
generating functions, similar to the one previously obtained in the orientable
case via the Marcus-Schaeffer bijection. It also shows that the renormalization
factor for distances between vertices is universal for maps on all
surfaces: the renormalized profile and radius in a uniform random pointed
bipartite quadrangulation on any fixed surface converge in distribution when
the size tends to infinity. Finally, we extend the Miermont and
Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our
construction opens the way to the study of Brownian surfaces for any compact
2-dimensional manifold.Comment: v2: 55 pages, 22 figure
A bijection for rooted maps on general surfaces (extended abstract)
International audienceWe extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent for both orientable and non-orientable maps of Euler characteristic and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor ¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage pour les cartes orientables et non-orientables de caractéristique d’Euler , et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation ¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte
A bijection for rooted maps on general surfaces (extended abstract)
We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent for both orientable and non-orientable maps of Euler characteristic and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor ¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold
Asymptotic enumeration of constellations and related families of maps on orientable surfaces
We perform the asymptotic enumeration of two classes of rooted maps on
orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they
correspond respectively to maps with even face degrees and bipartite maps. We
obtain explicit asymptotic formulas for the number of such maps with any finite
set of allowed face degrees.
Our proofs rely on the generalisation to orientable surfaces of the
Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We
show that each of the 2g fondamental cycles of the surface contributes a factor
m between the numbers of m-hypermaps and m-constellations -- for example, large
maps of genus g with even face degrees are bipartite with probability tending
to 1/2^{2g}.
A special case of our results implies former conjectures of Gao.Comment: 39 pages, 9 figure
A bijection for nonorientable general maps
We give a different presentation of a recent bijection due to Chapuy and
Dol\k{e}ga for nonorientable bipartite quadrangulations and we extend it to the
case of nonorientable general maps. This can be seen as a Bouttier--Di
Francesco--Guitter-like generalization of the Cori--Vauquelin--Schaeffer
bijection in the context of general nonorientable surfaces. In the particular
case of triangulations, the encoding objects take a particularly simple form
and this allows us to recover a famous asymptotic enumeration formula found by
Gao
Counting unicellular maps on non-orientable surfaces
A unicellular map is the embedding of a connected graph in a surface in such
a way that the complement of the graph is a topological disk. In this paper we
present a bijective link between unicellular maps on a non-orientable surface
and unicellular maps of a lower topological type, with distinguished vertices.
From that we obtain a recurrence equation that leads to (new) explicit counting
formulas for non-orientable unicellular maps of fixed topology. In particular,
we give exact formulas for the precubic case (all vertices of degree 1 or 3),
and asymptotic formulas for the general case, when the number of edges goes to
infinity. Our strategy is inspired by recent results obtained by the second
author for the orientable case, but significant novelties are introduced: in
particular we construct an involution which, in some sense, "averages" the
effects of non-orientability
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
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
Enumeration of N-rooted maps using quantum field theory
A one-to-one correspondence is proved between the N-rooted ribbon graphs, or
maps, with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum
field theory. This result is used to obtain explicit expressions and relations
for the generating functions of N-rooted maps and for the numbers of N-rooted
maps with a given number of edges using the path integral approach applied to
the corresponding quantum field theory.Comment: 27 pages, 7 figure
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