  Two results (together with their relatively elementary proofs) are presented.
  The first one presents the upper boundary on the number of spanning trees in
a finite planar multigraph, proving that the complexity (the number of spanning
trees) of a planar multigraph with $n$ edges does not exceed $\tau^n$, where
$\tau \approx 1.8637$.
  This result is, quite possibly, already known and/or published -- my quick
web search did not turn up anything but that does not really prove much. It
also seems plausible that this inequality is actually true for the "best
possible" value of $\tau^* \approx 1.7916$.
  The second result uses the above theorem to improve on the well-known
Conway's inequality for the number of tiles in a prime rectangular squaring.

Combinatorics