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Chapter 14: Problem 52

Describe how to determine the number of Hamilton circuits in a complete graph.

Short Answer

Expert verified
The number of Hamilton circuits in a complete graph can be determined using the formula \( (n-1)!/2 \), where n is the number of vertices in the graph. A Hamilton circuit is a path that visits each vertex exactly once and returns to the beginning, and this formula provides the total number of such circuits in the graph

Step by step solution

01

Count the Vertices

Determine the number of vertices (n) in the complete graph. A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.
02

Apply Hamilton Circuit Formula

Apply the formula for the number of Hamilton circuits in a complete graph, which is \( (n-1)!/2 \). The term \( (n-1)! \) is the factorial of the number of vertices minus one. The factorial of a number is the product of all positive integers less than or equal to that number. Then, divide by 2 because each circuit can be written in reverse order, creating duplicate count.

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