# Chapter 14 - Graph Theory - 14.3 Hamilton Paths and Hamilton Circuits - Exercise Set 14.3: 61

The original statement does not make sense.

#### Work Step by Step

The original statement does not make sense. The number of Hamilton circuits in a complete graph with $n$ vertices is $(n-1)!$. For example, if a complete graph has 6 vertices, then the number of Hamilton circuits is $(6-1)! = 5! = 120$. We have not yet been able to find a simple method to determine the optimal Hamilton circuit for a weighted graph. One method we can use is the Brute Force Method, which involves listing all the possible Hamilton circuits and calculating the total weight of each Hamilton circuit. For a graph with a small number of vertices, the Brute Force Method is a reasonable method. However, if a graph has a large number of vertices, the number of Hamilton circuits can be extremely large. Even for a super-computer, it would be impractical to list all the possible Hamilton circuits and calculate the total weight of each Hamilton circuit. If a complete graph has 20 vertices, then the number of Hamilton circuits is $(20-1)! = 19!$ which is an enormous number. Even for a super-computer, it would take much longer than one evening to list all the possible Hamilton circuits and calculate the total weight of each Hamilton circuit.

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.