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: 75

Answer

A,C,E,G,F,D,B,A,E,F,B,C,G,D,A,G,B,E,D,C,F,A is an Euler circuit.

Work Step by Step

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 connected graph has at least one Euler circuit. 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, AC, AD, AE, AF, and AG are not bridges, we can choose any of these edges as the next step in the path. For simplicity, first we can follow a path around the outside of the polygon. From vertex A, the path can travel to vertex C, then to vertex E, then to vertex G, then to vertex F, then to vertex D, then to vertex B, and then back to vertex A. According to Fleury's Algorithm, we should always choose an edge that is not a bridge, if possible. Since the edges AD, AE, AF, and AG are not bridges, we can choose any of these edges as the next step in the path. For simplicity, we can follow a path to every second vertex of the polygon. From vertex A, the path can travel to vertex E, then to vertex F, then to vertex B, then to vertex C, then to vertex G, then to vertex D, and then back to vertex A. According to Fleury's Algorithm, we should always choose an edge that is not a bridge, if possible. Since the edges AF and AG 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 G. After this step, the path must travel to vertex B, then to vertex E, then to vertex D, then to vertex C, then to vertex F, and finally back to vertex A, because these are the only available edges. This path is A,C,E,G,F,D,B,A,E,F,B,C,G,D,A,G,B,E,D,C,F,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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