203 research outputs found
Random planar maps and graphs with minimum degree two and three
We find precise asymptotic estimates for the number of planar maps and graphs
with a condition on the minimum degree, and properties of random graphs from
these classes. In particular we show that the size of the largest tree attached
to the core of a random planar graph is of order c log(n) for an explicit
constant c. These results provide new information on the structure of random
planar graphs.Comment: 32 page
Clusters, generating functions and asymptotics for consecutive patterns in permutations
We use the cluster method to enumerate permutations avoiding consecutive
patterns. We reprove and generalize in a unified way several known results and
obtain new ones, including some patterns of length 4 and 5, as well as some
infinite families of patterns of a given shape. By enumerating linear
extensions of certain posets, we find a differential equation satisfied by the
inverse of the exponential generating function counting occurrences of the
pattern. We prove that for a large class of patterns, this inverse is always an
entire function. We also complete the classification of consecutive patterns of
length up to 6 into equivalence classes, proving a conjecture of Nakamura.
Finally, we show that the monotone pattern asymptotically dominates (in the
sense that it is easiest to avoid) all non-overlapping patterns of the same
length, thus proving a conjecture of Elizalde and Noy for a positive fraction
of all patterns
A solution to the tennis ball problem
We present a complete solution to the so-called tennis ball problem, which is
equivalent to counting lattice paths in the plane that use North and East steps
and lie between certain boundaries. The solution takes the form of explicit
expressions for the corresponding generating functions. Our method is based on
the properties of Tutte polynomials of matroids associated to lattice paths. We
also show how the same method provides a solution to a wide generalization of
the problem.Comment: 9 pages, Late
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 probability of planarity of a random graph near the critical point
Consider the uniform random graph with vertices and edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
\lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
In this paper we determine the exact probability of a random graph being
planar near the critical point . For each , we find an exact
analytic expression for
In particular, we obtain .
We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of being
series-parallel converges to 0.98003.
For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur
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
Degree distribution in random planar graphs
We prove that for each , the probability that a root vertex in a
random planar graph has degree tends to a computable constant , so
that the expected number of vertices of degree is asymptotically ,
and moreover that .
The proof uses the tools developed by Gimenez and Noy in their solution to
the problem of the asymptotic enumeration of planar graphs, and is based on a
detailed analysis of the generating functions involved in counting planar
graphs. However, in order to keep track of the degree of the root, new
technical difficulties arise. We obtain explicit, although quite involved
expressions, for the coefficients in the singular expansions of the generating
functions of interest, which allow us to use transfer theorems in order to get
an explicit expression for the probability generating function . From this we can compute the to any degree of accuracy, and derive
the asymptotic estimate for large values of ,
where is a constant defined analytically
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