Suppose I have a simple, planar graph, which I want to partition into three edge sets such that each set forms a spanning tree. I've made an attempt at a solution, but it requires a few assumptions that I'm not sure I can make; here's what I've got so far:
Let's say we have a tree with n nodes; then that tree must have n-1 edges. [[ASSUMPTION: All possible spanning trees of a graph must have an equal number of nodes, and, those spanning trees must not be disjointed.]] Thus, given that the graph is both simple and planar, and assuming that we can partition the graph into three edge sets such that each set forms a spanning tree, then we can say that the graph has at most 3n-6 edges. Then if we create three equal partitions, each edge set can have at most n-2 edges, and there is no way to partition the graph so that ALL three sets have greater than n-2 edges, which is what we need to make every set a spanning tree of the graph (given that they must have n-1 edges).
I'm not quite sure though how I can make the leaps that: 1) All spanning trees in the graph must have a an equal number of nodes (though intuitively, this makes sense to me). 2) If a graph has multiple spanning sets within it, those sets are necessarily not disjointed. (I'm less intuitively sure of this one, and that's probably not a good way to approach the problem, but it seems possible, only I'm not sure how to prove it, but it's all I've got so far.)