222 research outputs found
Random cliques in random graphs
We show that for each , in a density range extending up to, and
slightly beyond, the threshold for a -factor, the copies of in the
random graph are randomly distributed, in the (one-sided) sense that
the hypergraph that they form contains a copy of a binomial random hypergraph
with almost exactly the right density. Thus, an asymptotically sharp bound for
the threshold in Shamir's hypergraph matching problem -- recently announced by
Jeff Kahn -- implies a corresponding bound for the threshold for to
contain a -factor. We also prove a slightly weaker result for , and
(weaker) generalizations replacing by certain other graphs . As an
application of the latter we find, up to a log factor, the threshold for
to contain an -factor when is -balanced but not strictly
-balanced.Comment: 19 pages; expanded introduction and Section 5, plus minor correction
The hitting time of rainbow connection number two
In a graph with a given edge colouring, a rainbow path is a path all of
whose edges have distinct colours. The minimum number of colours required to
colour the edges of so that every pair of vertices is joined by at least
one rainbow path is called the rainbow connection number of the graph
. For any graph , . We will show that for the
Erd\H{o}s-R\'enyi random graph close to the diameter 2 threshold, with
high probability if then . In fact, further strengthening
this result, we will show that in the random graph process, with high
probability the hitting times of diameter 2 and of rainbow connection number 2
coincide.Comment: 16 pages, 2 figure
Exploring hypergraphs with martingales
Recently, we adapted exploration and martingale arguments of Nachmias and
Peres, in turn based on ideas of Martin-L\"of, Karp and Aldous, to prove
asymptotic normality of the number of vertices in the largest component
of the random -uniform hypergraph throughout the supercritical regime.
In this paper we take these arguments further to prove two new results: strong
tail bounds on the distribution of , and joint asymptotic normality of
and the number of edges of . These results are used in a
separate paper "Counting connected hypergraphs via the probabilistic method" to
enumerate sparsely connected hypergraphs asymptotically.Comment: 32 pages; significantly expanded presentation. To appear in Random
Structures and Algorithm
Erratum: Percolation on random Johnson-Mehl tessellations and related models
We correct a simple error in Percolation on random Johnson-Mehl tessellations
and related models, Probability Theory and Related Fields 140 (2008), 417-468.
(See also arXiv:math/0610716)Comment: 7 page
A short proof of the Harris-Kesten Theorem
We give a short proof of the fundamental result that the critical probability
for bond percolation in the planar square lattice is equal to 1/2. The lower
bound was proved by Harris, who showed in 1960 that percolation does not occur
at . The other, more difficult, bound was proved by Kesten, who showed
in 1980 that percolation does occur for any .Comment: 17 pages, 9 figures; typos corrected. To appear in the Bulletin of
the London Mathematical Societ
Counting racks of order n
A rack on can be thought of as a set of maps , where
each is a permutation of such that
for all and . In 2013, Blackburn showed that the number of isomorphism
classes of racks on is at least and at most , where ; in this paper we improve the upper bound
to , matching the lower bound. The proof involves
considering racks as loopless, edge-coloured directed multigraphs on ,
where we have an edge of colour between and if and only if , and applying various combinatorial tools.Comment: Minor edits. 21 pages; 1 figur
An old approach to the giant component problem
In 1998, Molloy and Reed showed that, under suitable conditions, if a
sequence of degree sequences converges to a probability distribution , then
the size of the largest component in corresponding -vertex random graph is
asymptotically , where is a constant defined by the
solution to certain equations that can be interpreted as the survival
probability of a branching process associated to . There have been a number
of papers strengthening this result in various ways; here we prove a strong
form of the result (with exponential bounds on the probability of large
deviations) under minimal conditions.Comment: 24 pages; only minor change
Counting connected hypergraphs via the probabilistic method
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number
of connected graphs on with edges, whenever and the nullity
tend to infinity. Asymptotic formulae for the number of connected
-uniform hypergraphs on with edges and so nullity
were proved by Karo\'nski and \L uczak for the case ,
and Behrisch, Coja-Oghlan and Kang for . Here we prove such a
formula for any fixed, and any satisfying and
as . This leaves open only the (much simpler) case
, which we will consider in future work. ( arXiv:1511.04739 )
Our approach is probabilistic. Let denote the random -uniform
hypergraph on in which each edge is present independently with
probability . Let and be the numbers of vertices and edges in
the largest component of . We prove a local limit theorem giving an
asymptotic formula for the probability that and take any given pair
of values within the `typical' range, for any in the supercritical
regime, i.e., when where
and ; our enumerative result then follows
easily.
Taking as a starting point the recent joint central limit theorem for
and , we use smoothing techniques to show that `nearby' pairs of values
arise with about the same probability, leading to the local limit theorem.
Behrisch et al used similar ideas in a very different way, that does not seem
to work in our setting.
Independently, Sato and Wormald have recently proved the special case ,
with an additional restriction on . They use complementary, more enumerative
methods, which seem to have a more limited scope, but to give additional
information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical
changes. 67 pages (including appendix
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