## Thinking Mathematically (6th Edition)

In a complete graph with 4 vertices, the number of Hamilton circuits is $(4-1)! = 3! = 6$. There are 6 Hamilton circuits in the graph. The Brute Force Method involves listing the total weight of each Hamilton circuit and choosing the Hamilton circuit with the smallest weight. We can use the answers from Exercises 25-30 to find the optimal solution with the Brute Force Method. 25. The Hamilton circuit is A,B,C,D,A. The total weight is 88. 26. The Hamilton circuit is A,B,D,C,A. The total weight is 82. 27. The Hamilton circuit is A,C,B,D,A. The total weight is 70. 28. The Hamilton circuit is A,C,D,B,A. The total weight is 82. 29. The Hamilton circuit is A,D,B,C,A. The total weight is 70. 30. The Hamilton circuit is A,D,C,B,A. The total weight is 88. The optimal solutions are A,C,B,D,A and A,D,B,C,A. Both of these Hamilton circuits have a total weight of 70.