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Trees and matchings. (English) Zbl 0939.05066

Electron. J. Comb. 7, No. 1, Research paper R25, 34 p. (2000); printed version J. Comb. 7, No. 1 (2000).
Summary: 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 \(\mathcal H\). 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 R. Kenyon [Local statistics of lattice dimers, Ann. Inst. Henri PoincarĂ©, Probab. Stat. 33, No. 5, 591-618 (1997; Zbl 0893.60047)], 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.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C05 Trees

Citations:

Zbl 0893.60047
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Online Encyclopedia of Integer Sequences:

Number of spanning trees of quarter Aztec diamonds of order n.