Math 108 Topics in Mathematics | Laura Barnes |
2.1 Hamiltonian Circuits - Video

2.1 Hamiltonian Circuits - Video

Key Ideas

A Hamiltonian circuit of a graph visits each vertex exactly once, and returns to the starting point.
There is no simple way to determine if a graph has a Hamiltonian circuit, and it can be hard to construct one.

A number added to the edge of a graph is called a weight.
A minimum-cost Hamiltonian circuit is one with the lowest possible sum of the weights of its edges. A step-by-step process (called an algorithm) for finding a minimum-cost Hamilton circuit is to find all circuits, find the sum of the weights, and choose the tour with the minimum sum.

The method of trees can be used to help generate all possible Hamiltonian circuits.

The fundamental principle of counting says that if there are a ways of choosing one thing, b ways of choosing a second after the first, ..., and z ways of choosing the last item after the earlier choices then there are a total of axbx⋅⋅⋅x z total ways to make such choices.

Graphs may fail to have Hamiltonian circuits for a variety of reasons. One very important class of graphs, the complete graphs, automatically have Hamiltonian circuits. A graph is complete if every
pair of vertices is joined by an edge. If a complete graph has n vertices,
then there are (n−1)!/2 Hamiltonian circuits.

The brute force method is one algorithm that can be used to find a minimum-cost Hamiltonian circuit, but it is not a very practical method for a large problem. This method tries all possibilities.