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Published online by Cambridge University Press: 25 June 2025
This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and on-line literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems.
1. Introduction
How many perfect matchings does a given graph G have? That is, in how many ways can one choose a subset of the edges of G so that each vertex of G belongs to one and only one chosen edge? (See Figure l(a) for an example of a perfect matching of a graph.) For general graphs G, it is computationally hard to obtain the answer [Valiant 1979], and even when we have the answer, it is not so clear that we are any the wiser for knowing this number. However, for many infinite families of special graphs the number of perfect matchings is given by compellingly simple formulas. Over the past ten years a great many families of this kind have been discovered, and while there is no single unified result that encompasses all of them, many of these families resemble one another, both in terms of the form of the results and in terms of the methods that have been useful in proving them.
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