Spanning trees and orientations of graphs

A conjecture of Merino and Welsh says that the number of spanning trees τ(G) of a loopless and bridgeless multigraph G is always less than or equal to either the number a(G) of acyclic orientations, or the number c(G) of totally cyclic orientations, that is, orientations in which every edge is in a directed cycle. We prove that τ(G) ≤ c(G) if G has at least 4n edges, and that τ(G) ≤ a(G) if G has at most 16n/15 edges. We also prove that τ(G) ≤ a(G) for all multigraphs of maximum degree at most 3 and consequently τ(G) ≤ c(G) for any planar triangulation.