Thinking Mathematically (6th Edition)

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

Chapter 14 - Graph Theory - Chapter 14 Test: 12

Answer

(a) We can use a graph to model the streets of the neighborhood. (b) Since the graph has no Euler paths, there is no path that the police officer can follow to walk each street exactly once.
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Work Step by Step

(a) We can use a graph to model the streets of the neighborhood. Each edge represents one street and each vertex represents an intersection where two or more streets meet. (b) We need to verify the number of odd vertices. Vertex B, vertex C, vertex E, vertex H, vertex J, and vertex K are odd vertices. The graph has exactly six odd vertices. Therefore, the graph does not have any Euler paths. According to Euler's theorem, for a graph to have at least one Euler path, the graph must have either zero or exactly two odd vertices. Since the graph has no Euler paths, there is no path that the police officer can follow to walk each street exactly once.
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