1. 6.1 Hamilton Circuits and Hamilton Path
6.2: Complete Graph

2. A Hamilton path is a path that goes through each vertex of the graph once and only once.
F, A, B, E, C, G, D is a Hamilton path

3. A Hamilton circuit is a circuit that goes through each vertex of the graph once and only once (starting point and ending point is the same)
F, B, E, C, G, D, A, F is a Hamilton circuit

4. Example:
Identify Euler path, Euler circuit, Hamilton path, and/or Hamilton circuit
No Euler circuit or path
Hamilton path
Hamilton circuit
Euler Path
Hamilton Path

5. A complete graph with N vertices is a graph in which every pair of distinct vertices is joined by an edge. Symbol is KN
KN has N(N-1) / 2 edges
Examples:
K K4 K6
K3 has 3(2)/ (3)/2 = 6 edges (5)/2 = 15 edges
= 3 edges

6. The number of Hamilton circuits in KN is (N-1)! Example:
This complete graph has 4 vertices so there are
(4-1)! = 3! = 3·2 ·1= 6 Hamilton circuit
Let A be the reference point:
A, B, C, D, A
A, B, D, C, A
A, C, B, D, A
A, C, D, B, A
A, D, B, C, A
A, D, C, B, A
Review Factorials in class B A
Mirror
Image
(same circuit) C D

7. 6.3 Traveling Salesman Problems

8. Traveling Salesman problem is a real life problem that involves Hamilton circuits in complete graphs
Examples:
Routing school buses
Package deliveries
Scheduling jobs on a machine
Running errands around town
Traveling to many different destinations

9. A complete weighted graph is a complete graph with weights.
A weighted graph is a graph with numbers attached to its edges. These numbers are called weights.
A complete weighted graph is a complete graph with weights. 45 70 20

10. A business man has to travel to 4 different cities and return to his home town at the end of the trip. The weights of these edges are one-way airfares between any two cities. A reward is offered to anyone who can find him the cheapest trip.
Reward??? Hmm,
What is it?