Sedláček’s conjecture on minimal vertices for simple graphs

Ascertain whether α'(t) = o(log t), where α'(t) denotes the minimal number of vertices of a simple (not necessarily planar) graph with exactly t spanning trees.

Background

For simple graphs without planarity constraints, α'(t) is defined as the minimal number of vertices needed to realize exactly t spanning trees. Cayley’s formula yields the lower bound α'(t) = Ω(log t / log log t). Despite progress (e.g., Stong’s upper bounds in the planar case), Sedláček’s original conjecture for the nonplanar setting remains unresolved.