Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - 14.2 Euler Paths and Euler Circuits - Exercise Set 14.2: 48

Answer

This graph has four odd vertices, so according to Euler's theorem, there are no Euler paths for this graph. Therefore, it is not possible to establish a route for the security guard so that each street is walked exactly once.
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Work Step by Step

An Euler path is a path that travels through every edge of a graph exactly once. We need to determine if this graph has an Euler path. According to Euler's theorem, for a graph to have at least one Euler path, the number of odd vertices must be either 0 or 2. On this graph, vertex B, vertex D, vertex F, and vertex H are odd vertices. This graph has four odd vertices. Therefore, according to Euler's theorem, there are no Euler paths for this graph. Therefore, it is not possible to establish a route for the security guard so that each street is walked exactly once.
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