6.1 Hamilton Circuits and Hamilton Path

6.2: Complete Graph

• 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

• 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

Example:• Identify Euler path, Euler circuit, Hamilton

path, and/or Hamilton circuit

Euler Path

Hamilton Path

No Euler circuit or path

Hamilton path

Hamilton circuit

• 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:

K3 K4 K6

K3 has 3(2)/2 4(3)/2 = 6 edges 6(5)/2 = 15 edges

= 3 edges

• 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, AA, B, D, C, AA, C, B, D, AA, C, D, B, AA, D, B, C, AA, D, C, B, A

A B

CD

Mirror

Image

(same circuit)

Review Factorials in class

6.3 Traveling Salesman Problems

• 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

• 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.

4570

20

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?