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We introduce a class of graphs called compound graphs, generalizingrectangles, which are constructed out of copies of a planar bipartite basegraph. The main result is that the number of perfect matchings of everycompound graph is divisible by the number of matchings of its base graph. Ourapproach is to use Kasteleyn's theorem to prove a key lemma, from which thedivisibility theorem follows combinatorially. This theorem is then applied toprovide a proof of Problem 21 of Propp's Enumeration of Matchings, adivisibility property of rectangles. Finally, we present a new proof, in thesame spirit, of Ciucu's factorization theorem.

Generalizing the divisibility property of rectangle domino tilings