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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 (1-2-6-8-5-1). (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
- Graph Theory, Planar graph, Graph coloring, complete graph, Hamilton circuit