Thinking Mathematically (6th Edition)

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

Chapter 14 - Graph Theory - 14.3 Hamilton Paths and Hamilton Circuits - Concept and Vocabulary Check - Page 921: 6



Work Step by Step

Not every complete graph that has a Hamilton circuit has at least one Euler circuit. There are complete graphs with all odd vertices which have a Hamilton circuit. However, since the graph has odd vertices, the graph can not have an Euler circuit. A graph can only have an Euler circuit if all the vertices are even. For example, let's consider a complete graph with four vertices. Then the graph has (4-1)! Hamilton circuits (which is 6 Hamilton circuits). However, since all the vertices are odd, the graph has no Euler circuit.
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