  Ehrenborg noted that all tilings of a bipartite planar graph are encoded by
its cubical matching complex and claimed that this complex is collapsible. We
point out to an oversight in his proof and explain why these complexes can be
the union of collapsible complexes. Also, we prove that all links in these
complexes are suspensions up to homotopy. Furthermore, we extend the definition
of a cubical matching complex to planar graphs that are not necessarily
bipartite, and show that these complexes are either contractible or a disjoint
union of contractible complexes.
  For a simple connected region that can be tiled with dominoes ($2\times 1$
and $1\times 2$) and $2\times 2$ squares, let $f_i$ denote the number of
tilings with exactly $i$ squares. We prove that $f_0-f_1+f_2-f_3+\cdots=1$
(established by Ehrenborg) is the only linear relation for the numbers $f_i$.
