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: 6

Answer

False

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.
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.