MIT6_042JS10_lec17_sol - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 12 Prof. Albert R. Meyer revised March 13, 2010, 1024 minutes Solutions to In-Class Problems Week 6, Fri. Problem 1. Prove that a graph is a tree iff it has a unique simple path between any two vertices. Solution. Theorem 10.3.1 shows that in a tree there are unique simple paths between any two vertices. Conversely, suppose we have a graph, G , with unique paths. Now G is connected since there is a path between any two vertices. So we need only show that G has no simple cycles. But if there was a simple cycle in G , there are two paths between any two vertices on the cycle (going one way around the cycle or the other way around), a violation of uniqueness. So G must cannot have any simple cycles. Problem 2. The n-dimensional hypercube, H n , is a graph whose vertices are the binary strings of length n . Two vertices are adjacent if and only if they differ in exactly 1 bit. For example, in H 3 , vertices 111 and 011 are adjacent because they differ only in the first bit, while vertices 101 and 011 are not adjacent because they differ at both the first and second bits. (a) Prove that it is impossible to find two spanning trees of H 3 that do not share some edge. Solution. H 3 has 8 vertices so every spanning tree has 7 edges. But H 3 has only 12 edges, so any two sets of 7 edges must overlap. (b) Verify that for any two vertices x = y of H 3 , there are 3 paths from x to y in H 3 , such that, besides x and y , no two of those paths have a vertex in common....
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MIT6_042JS10_lec17_sol - Massachusetts Institute of...

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