The cubical matching complex revisited

Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed this is collapsible. We point out to an oversight in his proof explain why these complexes can be the disjoint union two or more collapsible complexes. also prove links suspensions up homotopy. Furthermore, we extend definition graphs not necessarily bipartite, show either contractible For simple connected region tiled with dominoes (2 × 1 2) 2 squares, let fi denote number exactly i squares. f0 − f1 + f2 f3 ⋯ = (established Ehrenborg) only linear relation for numbers fi.