Yet another question I was going over and struggled.
Given a 3-regular connected planar graph, so that every vertex lies on the edge of a face of length 4, of a face of length 6 and of a face of length 8. Find the number of vertices, edges and faces of each length.
Obviously, in order to solve it one will have to use:
But using just these three equations doesn't lead to any useful result. The best I managed to get out of them is:
And some identities connection the edges to number of vertices to number of faces - none enough to solve the question. It seems as if I miss some identity that lies with the fact that "each vertex lies on the edge of a face of length 4, of a face of length 6 and of a face of length 8" - it can't be that the number of each of these faces is equal, they seem to be certainly different actually. Yet $f_6$ disappears along the way and I don't know how to go on. Any help/direction/ideas will be extremely helpful!