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

TL;DR

This paper provides elementary proofs for bounds on the number of spanning trees in planar multigraphs and applies these results to improve bounds on the size of prime rectangular squarings.

Contribution

It introduces a new upper bound on the complexity of planar multigraphs and applies this to refine Conway's inequality for prime rectangular squarings.

Findings

Upper bound on spanning trees: τ^n with τ ≈ 1.8637

Potentially optimal bound: τ* ≈ 1.7916

Improved inequality for prime rectangular squaring sizes

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.