321 research outputs found
Spread-out percolation in R^d
Let be either or the points of a Poisson process in of
intensity 1. Given parameters and , join each pair of points of
within distance independently with probability . This is the simplest
case of a `spread-out' percolation model studied by Penrose, who showed that,
as , the average degree of the corresponding random graph at the
percolation threshold tends to 1, i.e., the percolation threshold and the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows immediately from of
a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated
references to [3]. To appear in Random Structures and Algorithm
Consistent random vertex-orderings of graphs
Given a hereditary graph property , consider distributions of
random orderings of vertices of graphs that are preserved
under isomorphisms and under taking induced subgraphs. We show that for many
properties the only such random orderings are uniform, and give
some examples of non-uniform orderings when they exist
Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs
The independence density of a finite hypergraph is the probability that a
subset of vertices, chosen uniformly at random contains no hyperedges.
Independence densities can be generalized to countable hypergraphs using
limits. We show that, in fact, every positive independence density of a
countably infinite hypergraph with hyperedges of bounded size is equal to the
independence density of some finite hypergraph whose hyperedges are no larger
than those in the infinite hypergraph. This answers a question of Bonato,
Brown, Kemkes, and Pra{\l}at about independence densities of graphs.
Furthermore, we show that for any , the set of independence densities of
hypergraphs with hyperedges of size at most is closed and contains no
infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page
Continuum percolation with steps in an annulus
Let A be the annulus in R^2 centered at the origin with inner and outer radii
r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a
Poisson process with intensity 1 and let G_A be the random graph with vertex
set {x_i} and edges x_ix_j whenever x_i-x_j\in A. We show that if the area of A
is large, then G_A almost surely has an infinite component. Moreover, if we fix
\epsilon, increase r and let n_c=n_c(\epsilon) be the area of A when this
infinite component appears, then n_c\to1 as \epsilon \to 0. This is in contrast
to the case of a ``square'' annulus where we show that n_c is bounded away from
1.Comment: Published at http://dx.doi.org/10.1214/105051604000000891 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
Minimal symmetric differences of lines in projective planes
Let q be an odd prime power and let f(r) be the minimum size of the symmetric
difference of r lines in the Desarguesian projective plane PG(2,q). We prove
some results about the function f(r), in particular showing that there exists a
constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.Comment: 16 pages + 2 pages of tables. This is a slightly revised version of
the previous one (Thm 6 has been improved, and a few points explained
Dense Subgraphs in Random Graphs
For a constant and a graph , let be
the largest integer for which there exists a -vertex subgraph of
with at least edges. We show that if then
is concentrated on a set of two integers. More
precisely, with
,
we show that is one of the two integers closest to
, with high probability.
While this situation parallels that of cliques in random graphs, a new
technique is required to handle the more complicated ways in which these
"quasi-cliques" may overlap
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