applied combinatorics, 4 th ed. alan tucker
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Applied Combinatorics, 4 th Ed. Alan Tucker. Section 2.2 Hamilton Circuits. Prepared by: Nathan Rounds and David Miller. Definitions. Hamilton Path – A path that visits each vertex in a graph exactly once. Possible Hamilton Path: A-F-E-D-B-C. F. F. A. B. B. D. D. C. C. E. E. - PowerPoint PPT PresentationTRANSCRIPT
04/21/23 Tucker, Sec. 2.2 1
Applied Combinatorics, 4th Ed.Alan Tucker
Section 2.2
Hamilton CircuitsPrepared by: Nathan Rounds and David Miller
04/21/23 Tucker, Sec. 2.2 2
Definitions
• Hamilton Path – A path that visits each vertex in a graph exactly once.
B
CD
E
F Possible Hamilton Path: A-F-E-D-B-CA B
CD
E
F
04/21/23 Tucker, Sec. 2.2 3
Definitions
• Hamilton Circuit – A circuit that visits each vertex in a graph exactly once.
Possible Hamilton Circuit: A-F-E-D-C-B-A
A B
CD
E
F
04/21/23 Tucker, Sec. 2.2 4
Rule 1
• If a vertex x has degree 2, both of the edges incident to x must be part of any Hamilton Circuit.
A B
CD
E
F Edges FE and ED must be included in a Hamilton Circuit if one exists.
04/21/23 Tucker, Sec. 2.2 5
Rule 2• No proper subcircuit, that is, a circuit not
containing all vertices, can be formed when building a Hamilton Circuit.
A B
CD
E
F Edges FE, FD, and DE cannot all be used in a Hamilton Circuit.
04/21/23 Tucker, Sec. 2.2 6
Rule 3• Once the Hamilton Circuit is required to use
two edges at a vertex x, all other (unused) edges incident at x can be deleted.
A B
CD
E
FIf edges FA and FE are required in a Hamilton Circuit, then edge FD can be deleted in the circuit building process.
04/21/23 Tucker, Sec. 2.2 7
Example
• Using rules to determine if either a Hamilton Path or a Hamilton Circuit exists.
AD
E
GF
I
H
CB
J K
04/21/23 Tucker, Sec. 2.2 8
Using Rules
• Rule 1 tells us that the red edges must be used in any Hamilton Circuit.
H
AD
E
GF
I
CB
JK
Vertices A and G are the only vertices of degree 2.
04/21/23 Tucker, Sec. 2.2 9
Using Rules
• Rules 3 and 1 advance the building of our Hamilton Circuit.
AD
E
GF
I
H
CB
J K
•Since the graph is symmetrical, it doesn’t matter whether we use edge IJ or edge IK.
•If we choose IJ, Rule 3 lets us eliminate IK making K a vertex of degree 2.
•By Rule 1 we must use HK and JK.
04/21/23 Tucker, Sec. 2.2 10
Using Rules
• All the rules advance the building of our Hamilton Circuit.
AD
E
GF
I
CB
J
H
K
Rule 2 allows us to eliminate edge EH and Rule 3 allows us to eliminate FJ. Now, according to Rule 1, we must use edges BF, FE, and CH.
04/21/23 Tucker, Sec. 2.2 11
Using Rules
• Rule 2 tells us that no Hamilton Circuit exists.
AD
E
GF
I
CB
J
Since the circuit A-C-H-K-J-I-G-E-F-B-A that we were forced to form does not include every vertex (missing D), it is a subcircuit. This violates Rule 2.
H
K
04/21/23 Tucker, Sec. 2.2 12
Theorem 1
• A graph with n vertices, n > 2, has a Hamilton circuit if the degree of each vertex is at least n/2.
A
B
C
DE
F
n = 6 n/2 = 3 Possible Hamilton Circuit: A-B-E-D-C-F-A
04/21/23 Tucker, Sec. 2.2 13
However, not “if and only if”
E
B
CD
FA B
CD
F Theorem 1 does not necessarily have to be true in order for a Hamilton Circuit to exist. Here, each vertex is of degree 2 which is less than n/2 and yet a Hamilton Circuit still exists.
04/21/23 Tucker, Sec. 2.2 14
Theorem 2
• Let G be a connected graph with n vertices, and let the vertices be indexed x1,x2,…,xn, so that deg(xi) deg(xi+1).
• If for each k n/2, either deg(xk) > k or deg(xn-k) n-k, then G has a Hamilton Circuit.
n/2 = 3
k = 3,2,or 1 Possible Hamilton Circuit: X1-X5-X3-X4-X2-X6-X1
X5
X1
X6
X3
X4
X2
04/21/23 Tucker, Sec. 2.2 15
Theorem 3
• Suppose a planar graph G, has a Hamilton Circuit H.
• Let G be drawn with any planar depiction.
• Let ri denote the number of regions inside the Hamilton Circuit bounded by i edges in this depiction.
• Let be the number of regions outside the circuit bounded by i edges. Then numbers ri and satisfy the following equation.
'ir
'ir
'( 2)( ) 0i ii
i r r
04/21/23 Tucker, Sec. 2.2 16
Use of Theorem 3'( 2)( ) 0i i
i
i r r 4
4 4
6 6
6
6
6
6
Planar Graph G
' '4 4 6 62( ) 4( ) 0r r r r
No matter where a Hamilton Circuit is drawn (if it exists), we know that and
. Therefore, and must have the same parity and
.
'4 4 3r r
'6 6 6r r r 'r
'4 4| | 3r r
04/21/23 Tucker, Sec. 2.2 17
Use of Theorem 3 Cont’d
'4 4 0r r
Eq. (*)
•Consider the case .
•This is impossible since then the equation would require that which is impossible since .
•We now know that , and therefore .
•Now we cannot satisfy Eq. (*) because regardless of what possible value is taken on by , it cannot compensate for the other term to make the equation equal zero.
•Therefore, no Hamilton Circuit can exist.
'6 6 0r r
'4 4 3r r
'6 6| | 2r r '
6 6| 4( ) | 8r r
'4 42( )r r
' '4 4 6 62( ) 4( ) 0r r r r
04/21/23 Tucker, Sec. 2.2 18
Theorem 4
• Every tournament has a directed Hamilton Path.• Tournament – A directed graph obtained from a
(undirected) complete graph, by giving a direction to each edge.
A B
C D
The tournaments (Hamilton Paths) in this graph are:A-D-B-C, B-C-A-D, C-A-D-B, D-B-C-A, and D-C-A-B.
(K4, with arrows)
04/21/23 Tucker, Sec. 2.2 19
Definition
• Grey Code uses binary sequences that are almost the same, differing in just one position for consecutive numbers.
A=000 B=100
C=110D=010
F=011G=111
H=101I=001
Advantages for using Grey Code:
-Very useful when plotting positions in space.
-Helps navigate the Hamilton Circuit code.
Example of an Hamilton Circuit:
000-100-110-010-011-111-101-001-000
04/21/23 Tucker, Sec. 2.2 20
Class Exercise• Find a Hamilton Circuit, or prove that one
doesn’t exist.
A B C
D E
F G H
Rule’s:
•If a vertex x has degree 2, both of the edges incident to x must be part of any Hamilton Circuit.
•No proper subcircuit, that is, a circuit not containing all vertices, can be formed when building a Hamilton Circuit.
•Once the Hamilton Circuit is required to use two edges at a vertex x, all other (unused) edges incident at x can be deleted.
04/21/23 Tucker, Sec. 2.2 21
Solution• By Rule One, the red edges must be used
• Since the red edges form subcircuits, Rule Two tells us that no Hamilton Circuits can exist.
A B C
D E
F G H