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 - Exercise Set 14.3 - Page 921: 12


(a) This graph must have Hamilton circuits because it is a complete graph. (b) The graph has 720 Hamilton circuits.

Work Step by Step

(a) This graph is a complete graph. Therefore, the graph must have Hamilton circuits. (b) This complete graph has 7 vertices. We know that the number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$. The number of Hamilton circuits in this graph is $(7-1)!$ which is $6!$. Since $6! = 720$, this graph has 720 Hamilton circuits.
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