The Cubical Matching Complex

Given a bipartite planar graph embedded in the plane, we define its cubical matching complex. By combining results of Kalai and Propp, we show that the cubical matching complex is collapsible. As a corollary, we obtain that a simply connected region R in the plane that can be tiled with lozenges and hexagons satisfies , where f (i) is the number of tilings with i hexagons. The same relation holds for a region tiled with dominoes and 2 x 2 squares. Furthermore, we show for a region that can be tiled with dominoes, that each link of the associated cubical complex is either collapsible or homotopy equivalent to a sphere.