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Unformatted text preview: AMS 301.3 (Fall, 2007) Estie Arkin Exam 1 – Solution sketch Mean 79.77, median 79, high 100 (5 of them!), low 33. 1. (14 points) Are the two graphs shown below isomorphic? If so, give the isomorphism; if not, give careful reasons for your answer. No. Graph on left is bipartite, graph on right is not (an odd circuit is 1-2-3-4-5-1). Also, graph on left is planar, graph on right is not. Note that both graphs have circuits of length 4, so an incorrect reason is - graph on left has circuits of length 4 but of graph on right does not (it does, such as 1-2-5-6-1). 2. (14 points) Compute the chromatic number (vertex colouring number) of the graph G shown below. Justify your answer! (Show a colouring with χ ( G ) colours (label each node with its colour), and argue that fewer colours cannot suffice.) A B C D E F G H I J K Example in class (text page 69) shows that 4 colours are needed for the subgraph on nodes F,G,H,I,J,K (wheel). In fact 4 colours are enough for the entire graph: Colour 1 at nodes A,C,G,J, colour 2 at nodes B,D,H, colour 3 at nodes E,I, and colour 4 at K....
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This note was uploaded on 08/08/2010 for the course AMS 301 taught by Professor Arkin during the Spring '08 term at SUNY Stony Brook.
- Spring '08