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: 7

Answer

(a) The graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph must have at least one Euler path. (b) A,B,E,D,C,A,D,B is an Euler path.

Work Step by Step

(a) Vertex A and vertex B are odd vertices. Vertex C, vertex D, and vertex E are even vertices. The graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph must have at least one Euler path. (b) If a graph has exactly two odd vertices, then any Euler path starts at one odd vertex and ends at the other odd vertex. Let's start at vertex A. Let's travel around the outside of the graph to vertex B, then to vertex E, then to vertex D, then to vertex B, and back to vertex A. There are only two edges which have not been used. We can travel to vertex D and then finally to vertex B. This path is A,B,E,D,C,A,D,B. This path travels through every edge of the graph exactly once, so it is an Euler path.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.