Transcript Hamilton Circuits Tucker 2
Applied Combinatorics, 4
th
Ed.
Alan Tucker
Section 2.2
Hamilton Circuits Prepared by: Nathan Rounds and David Miller 1 4/30/2020 Tucker, Sec. 2.2
Definitions
•
Hamilton Path
– A path that visits each vertex in a graph exactly once.
A Possible Hamilton Path: AFEDBC 4/30/2020 Tucker, Sec. 2.2
2
Definitions
•
Hamilton Circuit
– A circuit that visits each vertex in a graph exactly once.
F A B Possible Hamilton Circuit: AFEDCBA E 4/30/2020 D C Tucker, Sec. 2.2
3
Rule 1
• If a vertex
x
has degree 2, both of the edges incident to
x
must be part of any Hamilton Circuit.
F A B Edges FE and ED must be included in a Hamilton Circuit if one exists.
D E 4/30/2020 C Tucker, Sec. 2.2
4
Rule 2
• No proper subcircuit, that is, a circuit not containing all vertices, can be formed when building a Hamilton Circuit.
F A B Edges FE , FD , and DE cannot all be used in a Hamilton Circuit.
D E 4/30/2020 C Tucker, Sec. 2.2
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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.
F A B If edges FA and FE are required in a Hamilton Circuit, then edge FD can be deleted in the circuit building process.
D E 4/30/2020 C Tucker, Sec. 2.2
6
Example
• Using rules to determine if either a Hamilton Path or a Hamilton Circuit exists.
B D A C 4/30/2020 F J I E G Tucker, Sec. 2.2
H K 7
Using Rules
• Rule 1 tells us that the red edges must be used in any Hamilton Circuit.
B D A C Vertices A and G are the only vertices of degree 2.
E G H F I J 4/30/2020 K Tucker, Sec. 2.2
8
Using Rules
• Rules 3 and 1 advance the building of our Hamilton Circuit.
F B D A E G C H •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 .
I J 4/30/2020 K Tucker, Sec. 2.2
9
Using Rules
• All the rules advance the building of our Hamilton Circuit.
B D A C E 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 .
G F H I J 4/30/2020 K Tucker, Sec. 2.2
10
Using Rules
• Rule 2 tells us that no Hamilton Circuit exists.
B D A F E G C H Since the circuit ACHKJIGEFBA that we were forced to form does not include every vertex (missing D ), it is a subcircuit. This violates Rule 2.
I J 4/30/2020 K Tucker, Sec. 2.2
11
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 C B
n
= 6
n
/2 = 3 Possible Hamilton Circuit: A BEDCFA E F D 4/30/2020 Tucker, Sec. 2.2
12
E 4/30/2020
However, not “if and only if”
A 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.
Tucker, Sec. 2.2
13
Theorem 2
X 2 • Let
G
be a connected graph X 4 with
n
vertices, and let the vertices be indexed
x 1
,
x 2
,…,
x n ,
so that deg(
x i
deg(
x i+
1 ).
• If for each
k
deg(
x k
) > k then
G
n
/2, either or deg(
x n

k
has a Hamilton
n

k
, X 3 Circuit.
X 6 X 5
n
/2 = 3
k
= 3,2,or 1 Possible Hamilton Circuit: X 1 X 5 X 3 X 4 X 2 X 6 X 1 X 1 4/30/2020 Tucker, Sec. 2.2
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Theorem 3
• Suppose a planar graph
G
, has a Hamilton Circuit
H
.
• Let
G
be drawn with any planar depiction.
• Let
r i
denote the number of regions inside the Hamilton Circuit bounded by
i
edges in this depiction. bounded by
i
edges. Then numbers
r i
following equation.
r i
' and satisfy the
i
(
i
2)(
r i
r i
' ) 0 4/30/2020 Tucker, Sec. 2.2
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Planar Graph
G
4 6 4 6
Use of Theorem 3
i
(
i
2)(
r i
r i
' ) 0 6 2(
r
4
r
4 ' ) 4(
r
6
r
6 ' ) 0 6 4 6 6 No matter where a Hamilton Circuit is drawn (if it exists), we
r
4
r
4 3
r
6
r
6 6
r
 must have the same parity and
r
4
r
4  3 .
r
' 4/30/2020 Tucker, Sec. 2.2
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Use of Theorem 3 Cont’d
2(
r
4
r
4 ' ) 4(
r
6
r
6 ' ) 0 Eq. (*)
r
6 6 0
r
4
r
4 3 
r
6
r
6  2  4(
r
6
r
6 '
r
4 4 0 •Now we cannot satisfy Eq. (*) because regardless of what possible value is taken 2(
r
4
r
4 equation equal zero.
' ) •Therefore, no Hamilton Circuit can exist.
4/30/2020 Tucker, Sec. 2.2
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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 4/30/2020 C B The tournaments (Hamilton Paths) in this graph are: ADBC, BCAD, CADB, DBCA, and DCAB .
(K 4 , with arrows) D Tucker, Sec. 2.2
18
Definition
•
Grey Code
uses binary sequences that are almost the same, differing in just one position for consecutive numbers.
F=011 I=001 H=101 D=010 G=111 C=110 Advantages for using Grey Code: Very useful when plotting positions in space.
Helps navigate the Hamilton Circuit code.
Example of an Hamilton Circuit: 000100110010011111101001000 B=100 A=000 4/30/2020 Tucker, Sec. 2.2
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Class Exercise
• Find a Hamilton Circuit, or prove that one doesn’t exist.
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.
A F D B G E C H 4/30/2020 Tucker, Sec. 2.2
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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 H F G Tucker, Sec. 2.2
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