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 No Euler circuit or path Hamilton path Hamilton circuit Euler Path Hamilton Path

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, 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

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 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

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?