Generalizing the Divisibility Property of Rectangle Domino Tilings

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.