1,168 research outputs found
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
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