Does there exist a planar graph whose edges can be coloured either red, green or blue in such a way that the red edges form a spanning tree, the green edges form a spanning tree, and the blue edges form a spanning tree?
I know that a spanning tree must touch all nodes and this does not apply for a graph with 3 vertices, but I'm not quite sure how to go about proving that there does not exist.
Best Answer
Every triangulation decomposes into 3 edge-disjoint spanning trees, when the edges of one triangular face are doubled. See here.
However, if you do not double the edges such a coloring is impossible, since a planar graph has at most $3n-6$ edges (Euler's formula) and three disjoint spanning trees, need $3(n-1)$ edges.