A Complementation Theorem for Perfect Matchings of Graphs Having a Cellular Completion

Abstract

A cellular graph is a graph whose edges can be partitioned into 4-cycles (called cells) so that each vertex is contained in at most two cells. We present a "Complementation Theorem" for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of M. Ciucu (J. Algebraic Combin. 5 (1996), 87-103). As applications of the Complementation Theorem we obtain a new proof of Stanley's multivariate version of the Aztec diamond theorem, a weighted generalization of a result of Knuth (J. Algebraic Combin. 6 (1997), 253-257) concerning spanning trees of Aztec diamond graphs, a combinatorial proof of Yang's enumeration ("Three Enumeration Problems Concerning Aztec Diamonds," Ph.D. thesis, M.I.T., 1991) of matchings of fortress graphs and direct proofs for certain identities of Jockusch and Propp. © 1998 Academic Press.