9 research outputs found
Asymptotic enumeration of orientations
International audienceWe find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations
Asymptotic enumeration of orientations
We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3-orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c . n(-alpha)gamma(n), for suitable constants c, alpha, gamma with alpha = 4 for 2-orientations and alpha = 5 for 3-orientations. The proofs are based on singularity analysis of D-finite generating functions, using the Fuchsian theory of complex linear differential equations
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
The Geometry of Random Tournaments
A tournament is an orientation of a graph. Each edge is a match, directed towards the winner. The score sequence lists the number of wins by each team. In this article, by interpreting score sequences geometrically, we generalize and extend classical theorems of Landau (Bull. Math. Biophys. 15, 143–148 (1953)) and Moon (Pac. J. Math. 13, 1343–1345 (1963)), via the theory of zonotopal tilings
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
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. ([13]). On the other hand, Chudnovsky and Seymour ([8]) 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 n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγn, where c > 0 and γ ∼ 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (not 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
Asymptotics in the schema of simple varieties of trees
We study the functional equation y(x) = xA(y(x)) satisfied by the generating functions in the schema of simple varieties of trees. The radii of convergence r, R of y, A respectively satisfy y(r) <= R. In the subcritical case (y(r) < R), y_n behaves asymptotically as C·r^(-n)·n^(-3/2). In the critical case (y(r)=R), we approach the problem of determining the asymptotics of a_n when the information of y(x) is well known. To that end, we give sufficient conditions that ensure that A(z) can be extended to a delta domain around its dominant singularity R, which is needed to be able to apply the transfer theorem when determining the asymptotics. We do a similar analysis for the equation y(x) = x+ B(y(x)), which appears in the additive schema of simple varieties of trees. The general framework we propose includes the three examples of extension to a delta domain of the functions A(z) and B(z) encountered in the literature so far
Asymptotic enumeration of orientations of a graph as a function of the out-degree sequence
We prove an asymptotic formula for the number of orientations with given
out-degree (score) sequence for a graph . The graph is assumed to have
average degrees at least for some ,
and to have strong mixing properties, while the maximum imbalance (out-degree
minus in-degree) of the orientation should be not too large. Our enumeration
results have applications to the study of subdigraph occurrences in random
orientations with given imbalance sequence. As one step of our calculation, we
obtain new bounds for the maximum likelihood estimators for the Bradley-Terry
model of paired comparisons.Comment: Accepted to the Electronic Journal of Combinatoric