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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’11 : Mathematics for Computer Science March 30 Prof. Albert R Meyer revised Friday 8 th April, 2011, 08:50 Problem Set 7 Due : April 8 Reading: Chapter 11.7 – 11.11.3 , Coloring, Connectedness, & Trees; Chapter 12 , Planar Graphs; Chapter 14 , Sums and Asymptotics. Skip the following sections which will not be covered this term: Chapter 11.11.4 , Minimum Weight Span- ning Trees, Chapter 13 , State Machines, Chapter 14.6 , Double Sums, & Chapter 14.7.5 , Omega notation. Problem 1. (a) Give an example of a simple graph that has two vertices u ¤ v and two distinct paths between u and v , but no cycle including either u or v . Hint: There is an example with 5 vertices. (b) Prove that if there are different paths between two vertices in a simple graph, then the graph has a cycle. Problem 2. The entire field of graph theory began when Euler asked whether the seven bridges of K¨onigsberg could all be crossed exactly once. Abstractly, we can represent the parts of the city separated by rivers as vertices and the bridges as edges between the vertices. Then Euler’s question asks whether there is a closed walk through the graph that includes every edge in a graph exactly once. In his honor, such a walk is called an Euler tour . So how do you tell in general whether a graph has an Euler tour? At first glance this may seem like a daunting problem. The similar sounding problem of finding a cycle that touches every vertex exactly once is one of those Millenium Prize NP-complete problems known as the...
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This note was uploaded on 10/04/2011 for the course CSCI 101 taught by Professor Leighton during the Spring '11 term at MIT.
- Spring '11
- Computer Science