1. Hamiltonian Circuits and Paths Vocabulary
AQR Unit VII Networks and Graphs

2. What is a Hamiltonian path?
Objective: Use Hamiltonian graphs to solve problems such as scheduling or routing.
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.
What is a Hamiltonian path?
Trace a Hamiltonian path for the graph below.

3. What is a Hamiltonian circuit?
A Hamiltonian path which ends in the same place in which it began is called a Hamiltonian circuit.
What is a Hamiltonian circuit?
Trace a Hamiltonian circuit for graph below.
What is a weighted edge?
A weighted edge has a value assigned to the edge (i.e.- miles, time, gallons of gas, any unit of measure)

4. What is often called “mail man circuit”?
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.
Remember:
The path does not need to go over every edge but must pass through every vertex exactly once so no backtracking!

5. Summary and Review:
Euler Paths
Passes over edge exactly once.
May pass through a vertex more than once.
Think “Snow Plow” problem
Think “E” for “Edges” and “Euler”
Two odd degree vertices
Euler Circuits
Passes over edge exactly once & go back to starting vertex.
Zero odd degree vertices
Eulerize
Find the odd degrees and the shortest path b/t them
Add a copy of those edges along that shortest path
Now you have an Euler Circuit!

6. Summary and Review Continued:
Passes through every vertex exactly once.
Not necessarily over every edge.
Have to guess and check to see if it’s Hamiltonian. Degrees don’t matter.
Hamiltonian
Paths
Hamiltonian Circuits
Passes through every vertex exactly once and ends at the same vertex where it started.
Have to guess and check to find it.