The amount of spanning trees in a planar graph G is equal to the amount of spanning trees in the dual graph G*.
I would like to prove this, i know it's true, but i would like to show that it holds for every spanning tree in G that there exist one and only one co spanning tree in G* for the original spanning tree in G.
I made a drawing of the cubegraph that illustrates what i'm trying to prove
Here i have added a vertex inside every mask in G, to create the dual graph G*.
You can clearly see the idea. If you have a spanning tree in G, the co spanning tree is the edges not colored in G. But i would like to prove that this always count.
I have tried with a proof by bijection, but it is not making sense, and i would like someone to explain to me how i would go about this.