hw9 - you started(b If you do a walk on K n on average how...

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Math 55 - Homework 9 Due Monday, August 8, 2011 Directions: You should work on these problems and write up solutions in groups of two . Your answer should be clear enough that it explains to someone who does not already understand the answer why it works. (This is different than just convincing the grader that you understand.) 1. Suppose the G is a simple, undirected, planar graph and every vertex in G has degree at least 5. Prove that G has at least 12 vertices of degree exactly 5. 2. Suppose that T is a tree and e is an edge which goes between two vertices of T , but is not in T . Prove that if you add e to T , then the new graph has exactly one cycle (which doesn’t repeat a vertex). 3. Random Walks In a random walk, you start on one vertex of a graph, and randomly follow one edge to an adjacent vertex. All adjacent vertices are equally likely. (a) If you do a walk on K n , on average, how many steps does it take to get back where
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Unformatted text preview: you started? (b) If you do a walk on K n , on average, how many stepss does it take until you have visited every vertex at least once? (c) Find a simple, directed graph with 15 vertices, one picked as the starting vertex, so that on average it will take more than 5000 steps to get back to the start. 4. Connectivity Detective You are a detective hired to investigate an simple, undirected graph. You have a list of all n vertices. There are two special vertices s and t . You need to find out if there is a path from s to t . You start off knowing nothing about the edges. The only questions you can ask is if there is an edge between two vertices. How many questions do you need to answer for certain whether there is a path from s to t ? Prove that your answer is optimal....
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