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Unformatted text preview: AMS 301.3 (Fall, 2009) Estie Arkin Exam 1 – Solutions Mean 76.68, median 80, high 96, low 38, all scores out of 96 points. 1. (14 points) Are the two graphs shown below isomorphic? If so, give the isomorphism; if not, give careful reasons for your answer. No, they are not isomorphic. Graph on the left is bipartite, graph on the right is not, as it has an odd circuit (126851). (Or: graph on left is planar, right one is not. This requires proof.) 2. (14 points) Compute the chromatic number of the graph G shown below. Justify your answer! (Show a coloring with χ ( G ) colors (label each node with its color), and argue that fewer colors cannot suffice.) There are several odd circuits, the graph is not bipaprtite (A,B,C,D,E,A) so at least 3 colours are needed. There are many 3 colourings, such as: A=1, B=3, C=2, D=1, E=2, F=2, G=1, H=3, I=2, J=3. Common mistakes: Saying there is a K 5 or a 5 wheel, or that since the degree at each node is 3 then 3 colours are needed (which is true if colouring edges, not nodes!). 3. (20 points) True or False? If true, give a short proof. If false, give a counterexample: (a). Every complete bipartite graph K n,m is connected ( n,m ≥ 1). True. Every node “on the right” has an edge to “every node on the left”, so these are connected by a path of length 1. Also, every node on the right is connected by a path of length 2 to every other node on the right, going through a node, say node 1, on the left. Similarly, every node on the left is connected by a path of length 2 to every other node on the left. Common mistake: Saying that complete bipartite graphs have an edge between every pair of nodes....
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 Spring '08
 ARKIN
 Graph Theory, Planar graph, Graph coloring, complete graph, Hamilton circuit

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