Perfect matchings and spanning trees: squarishness, bijections and independence

Abstract: A number which is either the square of an integer or two times the square of an integer is called squarish. There are two main results in the literature on graphs whose number of perfect matchings is squarish: one due to Jockusch (for planar graphs invariant under rotation by 90 degrees) and the other due to the second author (concerning planar graphs with two perpendicular symmetry axes). We present in this paper a new such class, consisting of certain planar graphs that are only required to have one symmetry axis. Our proof relies on a natural bijection between the set of perfect matchings of two closely related (but not isomorphic!) families of graphs, which is interesting in its own right. The rephrasing of this bijection in terms of spanning trees turns out to be the most natural way to present this result. The basic move in the construction of the above bijection (which we call gliding) can also be used to extend Temperley's classical bijection between spanning trees of a planar graph and perfect matchings of a closely related graph. We present this, and as an application we answer an open question posed by Corteel, Huang and Krattenthaler. We also discuss another dimer bijection (used in the proof of the second author's result mentioned above), and deduce from a refinement of it new results for spanning trees. These include a finitary version of an independence result for the uniform spanning tree on $\Z^2$ due to Johansson, a counterpart of it, and a bijective proof of an independence result on edge inclusions in the uniform spanning tree on $\Z^2$ due to Lyons.