Logarithmic upper bound for the minimal vertices α(t) of planar simple graphs

Establish that the minimal number α(t) of vertices of a connected planar simple graph having exactly t spanning trees satisfies α(t) = O(log t) for all integers t ≥ 3.

Background

For t ≥ 3, the authors define α(t) as the smallest number of vertices of a connected planar simple graph with exactly t spanning trees. A general lower bound α(t) = Ω(log t) follows from an Euler characteristic argument bounding τ(G) exponentially in the number of vertices. Stong proved the current best general upper bound α(t) = O((log t)^{3/2}/(log log t)), motivating the conjectured matching logarithmic upper bound.