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: 8

Answer

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

Work Step by Step

(a) Vertex A and vertex C are odd vertices. Vertex B, vertex D, and vertex E are even vertices. The graph has exactly two odd vertices. Therefore, by Euler's theorem, the graph has 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 C. The path must travel from vertex C to vertex B. From there, let's travel around the outside of the rectangle to vertex A, then to vertex D, then to vertex E, and back to vertex B. There are only two edges which have not been used. The path can then travel to vertex E and then finally to vertex A. This path is C,B,A,D,E,B,E,A. This path travels through every edge of the graph exactly once, so it is an Euler path. This is one Euler path but there are other Euler paths in this graph also.
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