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Discrete structures 2

Question in the picture.Discrete structures 2.MATH 340 in McGill.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 4colors theorem is proven for all planar graphs with less than n verticesand that G is a graph with n vertices that contains the subgraph below. We have alreadyremoved the four vertices in the middle, resulting in a smaller planar graph, and let theinduction hypothesis give us a coloring of the six outer vertices. Now suppose that thiscoloring is like in the following ?gure… (a) Use the Kempechain argument to show that we can modify the coloring to end upin one of the two following con?gurations. (b) Show that in either case, we can conclude that G is 4-colorable. Hint: You may need to use another Kempechaz’n argument in one of the two cases…

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