6.1 Hamilton Circuits and Hamilton Path 6.2: Complete Graph.
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Transcript of 6.1 Hamilton Circuits and Hamilton Path 6.2: Complete Graph.
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6.1 Hamilton Circuits and Hamilton Path
6.2: Complete Graph
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• 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
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• 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
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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
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• 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
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• 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
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6.3 Traveling Salesman Problems
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• 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
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• 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
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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?