429 research outputs found
Percolation on sparse random graphs with given degree sequence
We study the two most common types of percolation process on a sparse random
graph with a given degree sequence. Namely, we examine first a bond percolation
process where the edges of the graph are retained with probability p and
afterwards we focus on site percolation where the vertices are retained with
probability p. We establish critical values for p above which a giant component
emerges in both cases. Moreover, we show that in fact these coincide. As a
special case, our results apply to power law random graphs. We obtain rigorous
proofs for formulas derived by several physicists for such graphs.Comment: 20 page
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Monotone properties of random geometric graphs have sharp thresholds
Random geometric graphs result from taking uniformly distributed points
in the unit cube, , and connecting two points if their Euclidean
distance is at most , for some prescribed . We show that monotone
properties for this class of graphs have sharp thresholds by reducing the
problem to bounding the bottleneck matching on two sets of points
distributed uniformly in . We present upper bounds on the threshold
width, and show that our bound is sharp for and at most a sublogarithmic
factor away for . Interestingly, the threshold width is much sharper for
random geometric graphs than for Bernoulli random graphs. Further, a random
geometric graph is shown to be a subgraph, with high probability, of another
independently drawn random geometric graph with a slightly larger radius; this
property is shown to have no analogue for Bernoulli random graphs.Comment: Published at http://dx.doi.org/10.1214/105051605000000575 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Weighted counting of solutions to sparse systems of equations
Given complex numbers , we define the weight of a
set of 0-1 vectors as the sum of over all
vectors in . We present an algorithm, which for a set
defined by a system of homogeneous linear equations with at most
variables per equation and at most equations per variable, computes
within relative error in time
provided for an absolute constant and all . A similar algorithm is constructed for computing
the weight of a linear code over . Applications include counting
weighted perfect matchings in hypergraphs, counting weighted graph
homomorphisms, computing weight enumerators of linear codes with sparse code
generating matrices, and computing the partition functions of the ferromagnetic
Potts model at low temperatures and of the hard-core model at high fugacity on
biregular bipartite graphs.Comment: The exposition is improved, a couple of inaccuracies correcte
Generating Random Elements of Finite Distributive Lattices
This survey article describes a method for choosing uniformly at random from
any finite set whose objects can be viewed as constituting a distributive
lattice. The method is based on ideas of the author and David Wilson for using
``coupling from the past'' to remove initialization bias from Monte Carlo
randomization. The article describes several applications to specific kinds of
combinatorial objects such as tilings, constrained lattice paths, and
alternating-sign matrices.Comment: 13 page
Extending a perfect matching to a Hamiltonian cycle
Graph TheoryInternational audienceRuskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property
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