Elementary proof for the bounds of the complexity of a planar multigraph and the size of a prime rectangular squaring

Abstract

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.