Thinking Mathematically (6th Edition)

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

Chapter 14 - Graph Theory - Chapter Summary, Review, and Test - Review Exercises: 14

Answer

a. Since the graph has zero odd vertices, the graph has at least one Euler circuit. b. The path A,B,C,D,B,E,D,F,C,A is an Euler circuit.

Work Step by Step

a. We need to verify the number of odd vertices in the graph. Since all the vertices in this graph are even, the graph has zero odd vertices. Therefore, this graph has at least one Euler circuit. b. To find an Euler circuit, we can start at any vertex. Let's start at vertex A. According to Fleury's Algorithm, we should always choose an edge that is not a bridge, if possible. Since the edges AB and AC are not bridges, we can choose either of these edges as the next step in the path. From vertex A, the path can travel to vertex B, then to vertex C, and then to vertex D. After this step, we can see that the edge DF is a bridge, so according to Fluery's Algorithm, we must choose a different edge. The path can then travel to vertex B. After this step, the path must travel to vertex E, then to vertex D, then to vertex F, then to vertex C, and then finally back to vertex A, because these are the only available edges. This path is A,B,C,D,B,E,D,F,C,A. This path travels through every edge of the graph exactly once, so it is an Euler path. Since it starts and ends at the same vertex, this path is an Euler circuit. This is one Euler circuit but there are other Euler circuits in this graph also.
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