1. Hamiltonian Circuit/Paths:
A Hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A Hamiltonian path does not necessarily pass through all the edges of the graph, however.
A Hamiltonian path which ends in the same place in which it began is called a Hamiltonian circuit.

2. Trace a Hamiltonian path
Only a Hamiltonian path, not a Hamiltonian circuit. The path did not end at the same vertex it started.
The path does not need to go over every edge but it can only go over an edge once and must pass through every vertex exactly once.
Hamiltonian Circuits are often called the mail man circuit because the mailman goes to every mailbox but does not need to go over every street.

3. N 6 5 2 3 4
Bank
Pharmacy
Post Office
Dry Cleaners
Grocery Store
Home Sweet Home
Video Store
Find the quickest route if you want to start and end at home and run all your errands (visit every place in this graph exactly once).
A weighted edge has a value assigned to the edge (i.e.- miles, time, gallons of gas, any unit of measure)

4. Review:
Euler Graphs
Passes over edge exactly once. May pass through a vertex more than once.
Hamiltonian
Graphs
Passes through every vertex exactly once but not necessarily over every edge.
The path ends at the same vertex it started.
Circuits