Generalizing the Divisibility Property of Rectangle Domino Tilings

Forest Tong

doi:10.37236/4520

Abstract

We introduce a class of graphs called compound graphs, which are constructed out of copies of a planar bipartite base graph, and explore the number of perfect matchings of compound graphs. The main result is that the number of matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's Enumeration of Matchings, a divisibility property of rectangles. Finally, we present a new proof, in the same spirit, of Ciucu's factorization theorem.

Highlights

We introduce a class of graphs called compound graphs, generalizing rectangles, which are constructed out of copies of a planar bipartite base graph
The main result is that the number of perfect matchings of every compound graph is divisible by the number of matchings of its base graph
The number of perfect matchings of a planar graph has been the subject of much study, and many striking numerical relations have been observed among the matchings of related graphs

Summary

Introduction

The number of perfect matchings of a planar graph (or equivalently, the number of domino tilings of its planar dual, as shown in Figure 1) has been the subject of much study, and many striking numerical relations have been observed among the matchings of related graphs. Ciucu’s factorization theorem provides a neat proof of the case #R(a, b) | #R(a, 2b + 1), and our divisibility result may be viewed as generalizing Ciucu’s theorem The similarity between these two theorems is not a coincidence; in the last section of the paper, the same linear-algebraic techniques used to prove the zero-sum lemma are applied to provide an alternative proof of Ciucu’s lemma when the graph is bipartite. Another related result is Kuo’s graphical condensation theorem, or more precisely, his series of graphical condensation relations presented in Section 2 of [6]. Neither 5-pillows, 7-pillows, nor 9-pillows (as defined in [2]) seem to share this divisibility property, Aztec diamonds do the electronic journal of combinatorics 23(3) (2016), #P3.5

Definition of Compound Graphs

Kasteleyn’s Theorem and Sign Functions

The Divisibility Theorem

Application to Rectangles

Ciucu’s Factorization Theorem