Generalizing the divisibility property of rectangle domino tilings

TL;DR

This paper introduces compound graphs, generalizing rectangles, and proves their perfect matchings are divisible by the base graph's matchings, extending known divisibility properties in graph tilings with new combinatorial proofs.

Contribution

It generalizes the divisibility property of rectangle domino tilings to a broader class of graphs called compound graphs, using Kasteleyn's theorem for the proof.

Findings

Proves divisibility of perfect matchings in compound graphs

Provides a new combinatorial proof of a known divisibility property

Extends the divisibility theorem to a broader class of graphs

Abstract

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. 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.