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final-review-solutions - 1 2 Graph coloring Find the...

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CS 173: Discrete Structures, Spring 2010 HW 12 These problems should not be turned in. They are to help you review for the final. 1. Planar graphs (a) Show that this graph is planar by redrawing it without crossings: A B C D E Solution: A B C D E (b) Suppose that G is an undirected connected simple planar graph with 10 vertices, all of degree 4. How many edges does it have? Use Euler’s formula and the Handshaking Theorem to calculate how many regions it has. Solution: The sum of the vertex degrees is 40. By the Handshaking Theorem, there must be half that many edges, i.e. 20 edges. Euler’s formula says that v - e + f = 2, where e is the number of edges, v the number of vertices, and f the number of regions. So, in our case, 10 - 20 + f = 2. So there must be 12 regions.
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Unformatted text preview: 1 2. Graph coloring Find the chromatic numbers for the following graphs. G1: A B C D E G2: 1 5 3 4 2 Solution: The chromatic number for G1 is 3 and the chromatic number for G2 is 4. The following pictures show that the two graphs can be colored with this many colors. But we still need to argue that a smaller set of colors won’t work. G1 graph contains a cycle of 3 nodes (A, B, and C). So coloring with two colors is hopeless. G2 contains a set of four nodes (1,2,3,4) for which all pairs are connected. So it can’t be colored with less than four colors. G1: A:red B:blue C:green D:red E:blue G2: 1:red 5:blue 3:green 4:yellow 2:blue 2...
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final-review-solutions - 1 2 Graph coloring Find the...

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