5,561 research outputs found
Counting coloured planar maps: differential equations
We address the enumeration of q-coloured planar maps counted bythe number of
edges and the number of monochromatic edges. We prove that the associated
generating function is differentially algebraic,that is, satisfies a
non-trivial polynomial differential equation withrespect to the edge variable.
We give explicitly a differential systemthat characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing a result of
Tutte dealing with their proper q-colourings. Instatistical physics terms, we
solvethe q-state Potts model on random planar lattices. This work follows a
first paper by the same authors, where the generating functionwas proved to be
algebraic for certain values of q,including q=1, 2 and 3. It isknown to be
transcendental in general. In contrast, our differential system holds for an
indeterminate q.For certain special cases of combinatorial interest (four
colours; properq-colourings; maps equipped with a spanning forest), we derive
from this system, in the case of triangulations, an explicit differential
equation of order 2 defining the generating function. For general planar maps,
we also obtain a differential equation of order 3 for the four-colour case and
for the self-dual Potts model.Comment: 43 p
Knot theory and matrix integrals
The large size limit of matrix integrals with quartic potential may be used
to count alternating links and tangles. The removal of redundancies amounts to
renormalizations of the potential. This extends into two directions: higher
genus and the counting of "virtual" links and tangles; and the counting of
"coloured" alternating links and tangles. We discuss the asymptotic behavior of
the number of tangles as the number of crossings goes to infinity.Comment: chapter of the book Random Matrix Theory, Eds Akemann, Baik and Di
Francesc
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
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
Three-dimensional maps and subgroup growth
In this paper we derive a generating series for the number of cellular
complexes known as pavings or three-dimensional maps, on darts, thus
solving an analogue of Tutte's problem in dimension three.
The generating series we derive also counts free subgroups of index in
via a simple bijection
between pavings and finite index subgroups which can be deduced from the action
of on the cosets of a given subgroup. We then show that this
generating series is non-holonomic. Furthermore, we provide and study the
generating series for isomorphism classes of pavings, which correspond to
conjugacy classes of free subgroups of finite index in .
Computational experiments performed with software designed by the authors
provide some statistics about the topology and combinatorics of pavings on
darts.Comment: 17 pages, 6 figures, 1 table; computational experiments added; a new
set of author
Enumeration of Hypermaps of a Given Genus
This paper addresses the enumeration of rooted and unrooted hypermaps of a
given genus. For rooted hypermaps the enumeration method consists of
considering the more general family of multirooted hypermaps, in which darts
other than the root dart are distinguished. We give functional equations for
the generating series counting multirooted hypermaps of a given genus by number
of darts, vertices, edges, faces and the degrees of the vertices containing the
distinguished darts. We solve these equations to get parametric expressions of
the generating functions of rooted hypermaps of low genus. We also count
unrooted hypermaps of given genus by number of darts, vertices, hyperedges and
faces.Comment: 42 page
Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes
A quadrangulation is a graph embedded on the sphere such that each face is
bounded by a walk of length 4, parallel edges allowed. All quadrangulations can
be generated by a sequence of graph operations called vertex splitting,
starting from the path P_2 of length 2. We define the degree D of a splitting S
and consider restricted splittings S_{i,j} with i <= D <= j. It is known that
S_{2,3} generate all simple quadrangulations.
Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}.
First we show that the splittings S_{1,2} are exactly the monotone ones in the
sense that the resulting graph contains the original as a subgraph. Then we
show that they define a set of nontrivial ancestors beyond P_2 and each
quadrangulation has a unique ancestor.
Our results have a direct geometric interpretation in the context of
mechanical equilibria of convex bodies. The topology of the equilibria
corresponds to a 2-coloured quadrangulation with independent set sizes s, u.
The numbers s, u identify the primary equilibrium class associated with the
body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate
all primary classes from a finite set of ancestors which is closely related to
their geometric results.
If, beyond s and u, the full topology of the quadrangulation is considered,
we arrive at the more refined secondary equilibrium classes. As Domokos,
L\'angi and Szab\'o showed recently, one can create the geometric counterparts
of unrestricted splittings to generate all secondary classes. Our results show
that S_{1,2} can only generate a limited range of secondary classes from the
same ancestor. The geometric interpretation of the additional ancestors defined
by monotone splittings shows that minimal polyhedra play a key role in this
process. We also present computational results on the number of secondary
classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table
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