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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 Fall '11
 PatriciaBishop
 Graph Theory, Glossary of graph theory, Hamiltonian path, Hamilton circuit

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