How can i prove that every planar graph can be expressed as a union of five edge-disjoint forests ?
I think I should use theorem that says : ' Every planar graph contains vertex with degree 5 or less' and maybe continue similalry to five color theorem, but I am not sure.
Any hints are welcomed.
Thanks for help.
EDIT
I narrowed it to this :
We know that every planar graph contains vertex of degree <=5. If we pull of this vertex with its adjacent edges, we will get a smaller graph with maximum of 5 edges, which we can easily distribute into 5 forests(or color into 5 different colors for easier imagination). We just repeat the process until we color the whole graph. Now we need to prove that there won't be any cyrcles in forests(triangles(or bigger) with all edges of the same color). I think it should be done by contradiction. If we assume that there is a cyrcle in one of the forests, there should be something that contradicts the algorithm. (When I tried some examples on paper, I found out that there were cases where 2/3 of a triangle was the same color, but never a full cyrcle.)
If you could tell me how to finish this, I would be very grateful.
As you say, "Every planar graph contains vertex with degree 5 or less". I think you can use this together with induction on the number of vertices to get what you want: pull off the vertex of degree <= 5 and see what you can do.