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Traveling-Salesman Problem. Ch. 6. Hamilton Circuits. Euler circuit/path => Visit each edge once and only once Hamilton circuit => Visit each vertex once and only once (except at the end, where it returns to the starting vertex) Hamilton path => Visit each vertex once and only once

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hamilton circuits
Hamilton Circuits
  • Euler circuit/path => Visit each edge once and only once
  • Hamilton circuit => Visit each vertex once and only once (except at the end, where it returns to the starting vertex)
  • Hamilton path => Visit each vertex once and only once
  • Difference: Edge (Euler)  Vertex (Hamilton)
examples of hamilton circuits

A

B

E

D

C

Examples of Hamilton circuits
  • Has many Hamilton circuits:
  • A, B, C, D, E, A
  • A, D, C, E, B, A
  • Has many Hamilton paths:
  • A, B, C, D, E
  • A, D, C, E, B
  • Has no Euler circuit, no Euler path => 4 vertices of odd degree

Graph 1

Hamilton circuits can be shortened into a Hamilton path

by removal of the last edge

examples of hamilton circuits4
Examples of Hamilton circuits

A

B

  • Has no Hamilton circuits:
  • What ever the starting point, we are going to have to pass through vertex E more than once to close the circuit.
  • Has many Hamilton paths:
  • A, B, E, C, D
  • C, D, E, A, B
  • Has Euler circuit => each vertex has even degree

E

D

C

Graph 2

examples of hamilton circuits5
Examples of Hamilton circuits

F

A

B

  • Has many Hamilton circuits:
  • A, F, B, E, C, G, D, A
  • A, F, B, C, G, D, E, A
  • Has many Hamilton paths:
  • A, F, B, E, C, G, D
  • A, F, B, C, G, D, E
  • Has Euler circuit => Every vertex has even degree

E

D

C

G

Graph 3

examples of hamilton circuits6
Examples of Hamilton circuits

G

F

Has no Hamilton circuits:

Has no Hamilton paths:

Has no Euler circuit

Has no Euler path => more than 2 vertices of odd degree

A

B

E

D

C

I

H

Graph 4

complete graph

A

B

D

C

Complete graph
  • A graph with N vertices in which every pair of vertices is joined by exactly one edge is called the complete graph.
  • Total no. of edges = N(N-1)/2

In K4, each vertex

has degree 3 and

the number of

edges = 4 (3)/2 = 6

the six hamilton circuits of k 4
The six Hamilton circuits of K4

A

B

D

C

Rows => 6 Hamilton circuits

Cols=> same Hamilton circuit with different reference points

Graph

Reference point A

Reference point B

Reference point C

Reference point D

complete graph9
Complete graph
  • The number of Hamilton circuits in a complete graph can be computed by using factorials.
  • N! (factorial of N) = 1x 2x3x4x … x(N-1)x N
  • The complete graph with N vertices has

(N-1)! Hamilton circuits.

  • Example: The complete graph with 5 vertices has 4! = 1x2x3x4 = 24 Hamilton circuits
factorial
Factorial

Which of the following is true?

n! = n! x (n-1)!

n! = n! + (n-1)!

n! = n x (n-1)!

n! = n + (n-1)!

no of edges
No. of edges

No of edges in K10 is

  • 10
  • 10!
  • 90
  • 45
complete graph12
Complete graph

In a complete graph with 14 vertices (A through N), the total number of Hamilton circuits (including mirror-image circuits) that start at vertex A is 

  • 14!
  • (14x13)/2
  • 15!
  • 13!