36,790 research outputs found
The number of maximum matchings in a tree
We determine upper and lower bounds for the number of maximum matchings
(i.e., matchings of maximum cardinality) of a tree of given order.
While the trees that attain the lower bound are easily characterised, the trees
with largest number of maximum matchings show a very subtle structure. We give
a complete characterisation of these trees and derive that the number of
maximum matchings in a tree of order is at most (the
precise constant being an algebraic number of degree 14). As a corollary, we
improve on a recent result by G\'orska and Skupie\'n on the number of maximal
matchings (maximal with respect to set inclusion).Comment: 38 page
Invasion percolation on the Poisson-weighted infinite tree
We study invasion percolation on Aldous' Poisson-weighted infinite tree, and
derive two distinct Markovian representations of the resulting process. One of
these is the limit of a representation discovered by Angel et
al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration
process of a randomly weighted Poisson incipient infinite cluster. The dynamics
of the new process are much more straightforward to describe than those of
invasion percolation, but it turns out that the two processes have extremely
similar behavior. Finally, we introduce two new "stationary" representations of
the Poisson incipient infinite cluster as random graphs on which
are, in particular, factors of a homogeneous Poisson point process on the upper
half-plane .Comment: Published in at http://dx.doi.org/10.1214/11-AAP761 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Clique trees of infinite locally finite chordal graphs
We investigate clique trees of infinite locally finite chordal graphs. Our
main contribution is a bijection between the set of clique trees and the
product of local finite families of finite trees. Even more, the edges of a
clique tree are in bijection with the edges of the corresponding collection of
finite trees. This allows us to enumerate the clique trees of a chordal graph
and extend various classic characterisations of clique trees to the infinite
setting
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