### Generalizing the Divisibility Property of Rectangle Domino Tilings

#### 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.