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: 49

Answer

(a) please see graph below (b) Since this graph has at least one Euler path, we can establish a route so that the security guard will be able to walk each street exactly once. The security guard must start walking either at vertex B or vertex E.
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Work Step by Step

(a) please see graph below (b) 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, and vertex E are odd vertices. The other vertices are even vertices. This graph has exactly two odd vertices. Therefore, according to Euler's theorem, this graph has an Euler path. An Euler path is a path that travels through each edge on the graph exactly once. Since this graph has at least one Euler path, the security guard will be able to walk each street exactly once. When a graph has exactly two odd vertices, any Euler path starts at one odd vertex and ends at the other odd vertex. The security guard must start walking either at vertex B or vertex E.
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