Question 2 (30 marks) The goal of this problem is to prove one of the multiple cases of the 4-colors theorem. Let our four colors be: red, blue, green and yellow. Assume that the 4colors theorem is proven for all planar graphs with less than n vertices and that G' is a graph with n vertices that contains the subgraph below. We have already removed the four vertices in the middle, resulting in a smaller planar graph, and let the induction hypothesis give us a coloring of the six outer vertices. Now suppose that this coloring is like in the following gure... (a) Use the Kempechain argument to show that we can modify the coloring to end up in one of the two following congurations. (b) Show that in either case, we can conclude that G' is 4-colorable. Hint: You may need to use another Kempechaz'n argument in one of the two cases