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15.3 Hamilton Paths &amp; Circuits

# 15.3 Hamilton Paths &amp; Circuits - 15.3 Hamilton...

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15.3 Hamilton Paths and Hamilton Circuits A path that passes through each vertex of a graph exactly once is called a Hamilton path . If a Hamilton path begins and ends at the same vertex, it is called a Hamilton circuit . A complete graph has an edge between each pair of vertices. Every complete graph with three or more vertices must have a Hamilton circuit. The number of Hamilton circuits in a complete graph with n vertices is (n – 1)! . A complete graph whose edges have numbers attached to them is called a weighted graph ; those numbers are called the weights of the edges. The traveling sales person problem is the problem of finding a Hamilton circuit in a complete weighted graph for which the sum of the weights is a minimum; such a circuit is called the optimal Hamilton circuit or the optimal solution . The brute force method of solving this type of problem would be to [1] model the problem with a complete, weighted graph; [2] list all possible Hamilton circuits; [3] find the sum of each of the circuits; and [4] choose the optimal solution.
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