7 research outputs found
A note on correlations in randomly oriented graphs
Given a graph , we consider the model where is given a random
orientation by giving each edge a random direction. It is proven that for
, the events and are positively
correlated. This correlation persists, perhaps unexpectedly, also if we first
condition on \{s\nto t\} for any vertex . With this conditioning it
is also true that and are negatively correlated.
A concept of increasing events in random orientations is defined and a
general inequality corresponding to Harris inequality is given.
The results are obtained by combining a very useful lemma by Colin McDiarmid
which relates random orientations with edge percolation, with results by van
den Berg, H\"aggstr\"om, Kahn on correlation inequalities for edge percolation.
The results are true also for another model of randomly directed graphs.Comment: 7 pages. The main lemma was first published by Colin McDiarmid.
Relevant reference added and text rewritten to reflect this fac
On percolation and the bunkbed conjecture
We study a problem on edge percolation on product graphs . Here
is any finite graph and consists of two vertices connected
by an edge. Every edge in is present with probability
independent of other edges. The Bunkbed conjecture states that for all and
the probability that is in the same component as is greater
than or equal to the probability that is in the same component as
for every pair of vertices .
We generalize this conjecture and formulate and prove similar statements for
randomly directed graphs. The methods lead to a proof of the original
conjecture for special classes of graphs , in particular outerplanar graphs.Comment: 13 pages, improved exposition thanks to anonymous referee. To appear
in CP
Correlations for paths in random orientations of G(n,p) and G(n,m)
We study random graphs, both and , with random orientations
on the edges. For three fixed distinct vertices s,a,b we study the correlation,
in the combined probability space, of the events a -> s and s -> b.
For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p<p_c the
correlation is negative for large enough n and for p>p_c the correlation is
positive for large enough n. We conjecture that for a fixed n\ge 27 the
correlation changes sign three times for three critical values of p.
For G(n,m) it is similarly proved that, with , there is a
critical p_c that is the solution to a certain equation and approximately equal
to 0.7993. A lemma, which computes the probability of non existence of any k
directed edges in G(n,m), is thought to be of independent interest.
We present exact recursions to compute P(a -> s). We
also briefly discuss the corresponding question in the quenched version of the
problem.Comment: Author added, main proof greatly simplified and extended to cover
also G(n,m). Discussion on quenched version adde