4,012 research outputs found
Planar maps and continued fractions
We present an unexpected connection between two map enumeration problems. The
first one consists in counting planar maps with a boundary of prescribed
length. The second one consists in counting planar maps with two points at a
prescribed distance. We show that, in the general class of maps with controlled
face degrees, the solution for both problems is actually encoded into the same
quantity, respectively via its power series expansion and its continued
fraction expansion. We then use known techniques for tackling the first problem
in order to solve the second. This novel viewpoint provides a constructive
approach for computing the so-called distance-dependent two-point function of
general planar maps. We prove and extend some previously predicted exact
formulas, which we identify in terms of particular Schur functions.Comment: 47 pages, 17 figures, final version (very minor changes since v2
Restricted non-separable planar maps and some pattern avoiding permutations
Tutte founded the theory of enumeration of planar maps in a series of papers
in the 1960s. Rooted non-separable planar maps are in bijection with
West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori,
Jacquard and Schaeffer in 1997, as well as a family of permutations defined by
the avoidance of two four letter patterns. In this paper we give upper and
lower bounds on the number of multiple-edge-free rooted non-separable planar
maps. We also use the bijection between rooted non-separable planar maps and a
certain class of permutations, found by Claesson, Kitaev and Steingrimsson in
2009, to show that the number of 2-faces (excluding the root-face) in a map
equals the number of occurrences of a certain mesh pattern in the permutations.
We further show that this number is also the number of nodes in the
corresponding beta(1,0)-tree that are single children with maximum label.
Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure
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
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte has described in the book "Connectivity in graphs" a canonical
decomposition of any graph into 3-connected components. In this article we
translate (using the language of symbolic combinatorics)
Tutte's decomposition into a general grammar expressing any family of graphs
(with some stability conditions) in terms of the 3-connected subfamily. A key
ingredient we use is an extension of the so-called dissymmetry theorem, which
yields negative signs in the grammar.
As a main application we recover in a purely combinatorial way the analytic
expression found by Gim\'enez and Noy for the series counting labelled planar
graphs (such an expression is crucial to do asymptotic enumeration and to
obtain limit laws of various parameters on random planar graphs). Besides the
grammar, an important ingredient of our method is a recent bijective
construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
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
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof
We consider the problem of enumeration of planar maps and revisit its
one-matrix model solution in the light of recent combinatorial techniques
involving conjugated trees. We adapt and generalize these techniques so as to
give an alternative and purely combinatorial solution to the problem of
counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
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