417 research outputs found
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
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
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
Connectivity for bridge-addable monotone graph classes
A class A of labelled graphs is bridge-addable if for all graphs G in A and
all vertices u and v in distinct connected components of G, the graph obtained
by adding an edge between u and u is also in A; the class A is monotone if for
all G in A and all subgraphs H of G, H is also in A. We show that for any
bridge-addable, monotone class A whose elements have vertex set 1,...,n, the
probability that a uniformly random element of A is connected is at least
(1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This
establishes the special case of a conjecture of McDiarmid, Steger and Welsh
when the condition of monotonicity is added. This result has also been obtained
independently by Kang and Panagiotiou (2011).Comment: 11 page
The enumeration of planar graphs via Wick's theorem
A seminal technique of theoretical physics called Wick's theorem interprets
the Gaussian matrix integral of the products of the trace of powers of
Hermitian matrices as the number of labelled maps with a given degree sequence,
sorted by their Euler characteristics. This leads to the map enumeration
results analogous to those obtained by combinatorial methods. In this paper we
show that the enumeration of the graphs embeddable on a given 2-dimensional
surface (a main research topic of contemporary enumerative combinatorics) can
also be formulated as the Gaussian matrix integral of an ice-type partition
function. Some of the most puzzling conjectures of discrete mathematics are
related to the notion of the cycle double cover. We express the number of the
graphs with a fixed directed cycle double cover as the Gaussian matrix integral
of an Ihara-Selberg-type function.Comment: 23 pages, 2 figure
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