Math 310 fall 2003 combinatorial problem solving lecture 6 friday september 12
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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 6, Friday, September 12 PowerPoint PPT Presentation


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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 6, Friday, September 12. 2.2 Hamilton Circuits. Homework (MATH 310#2F): Read 2.3. Write down a list of all newly introduced terms (printed in boldface or italic) Do Exercises 2.2: 2,4a-g,16,20 Volunteers: ____________ ____________

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 6, Friday, September 12

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Math 310 fall 2003 combinatorial problem solving lecture 6 friday september 12

MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 6, Friday, September 12


2 2 hamilton circuits

2.2 Hamilton Circuits

  • Homework (MATH 310#2F):

    • Read 2.3. Write down a list of all newly introduced terms (printed in boldface or italic)

    • Do Exercises 2.2: 2,4a-g,16,20

    • Volunteers:

      • ____________

      • ____________

      • Problem: 16.

  • On Monday you will also turn in the list of all new terms with the following marks

    • + if you think you do not need the definition on your cheat sheet,

    • check (if you need just the term as a reminder),

    • - if you need more than just the definition to understand the term.


  • Hamilton circuits and paths

    Hamilton Circuits and Paths

    • A ciruit that visits every vertex of a graph is called a Hamilton circuit.

    • A path that visits every vertex of a graph is called a Hamilton path.


    The three rules

    The Three Rules

    • Rule1. If a vertex x has degree 2, both edges incident to x must be part of any Hamilton circuit.

    • Rule 2. No proper subcircuit can be formed when building a Hamilton circuit.

    • Rule 3. Once the Hamilton circuit is required to use two edges at a vertex x, all other edges incident to x must be removed from consideration.


    Example 1

    Example 1

    • Show that the graph on the left has no Hamilton circuit.

    • Hint: Apply Rule 1 four times.


    Example 2

    Example 2

    • Show that the graph on the left has no Hamilton circuit.

    • Hint: Apply Rule 1 twice and use symmetry.


    Example 3

    Example 3

    • Show that the graph on the left has no Hamilton circuit.

    • Hint: Apply Rule 1 twice and use symmetry.

    click


    A useful result not from the textbook

    A Useful Result (not from the textbook)

    • If a connected graph G contains k vertices whose removal disconnects G into more than k pieces, then G has no Hamilton circuit.


    Theorem 1 dirac 1952

    Theorem 1 (Dirac, 1952)

    • A graph with n vertices, n > 2, has a Hamilton circuit if the degree of each vertex is at least n/2.


    Theorem 2 chvatal 1972

    Theorem 2 (Chvatal, 1972)

    • Let G be a connected graph with n vertices: x1, x2, ..., xn, so that deg(x1) · deg(x2) · ... · deg(xn). If for each k · n/2, either deg(xk) > k or deg(xn-k) ¸ n – k, then G has a Hamilton circuit.


    Theorem 3 grinberg 1968

    Theorem 3 (Grinberg, 1968)

    • Suppose a plane graph G has a Hamilton circuit H. Let ri denote the number of regions inside the Hamilton circuit bounded by i edges. Let r’i be the number of regions outside the circuit bounded by i edges. Then the numbers ri and r’i satisfy the equation:

      • (3 - 2)(r3 – r’3) + (4 - 2)(r4 – r’4) + (5 - 2)(r5 – r’5) + ... = 0


    Example 4

    Example 4

    • Here we have:

      • (a) r4 + r’4 = 3

      • (b) r6 + r’6 = 6

    • By Grinberg we should also have:

      • (c) 2(r4 – r’4) + 4(r6 – r’6) = 0.

    • r4¹ r’4) r6¹ r’6.

    • Hence |r6 – r’6| ¸ 2.

    • |r6 – r’6| ¸ 2 \implies |r4 – r’4| ¸ 4.

    • Contradiction! The graph in Figure 2.8 has no Hamilton circuit.


    Tournament

    Tournament

    • A tournament is a directed graph obtained from a complete graph by giving a direction to each edge.


    Theorem 4

    Theorem 4

    • Every tournament has a (directed) Hamilton path.

    • Proof. By induction on the number of vertices.


    Example 5 gray code

    Example: n = 3. There are 8 binary sequences:

    000

    001

    010

    011

    100

    101

    110

    111

    There are 2n binary sequences of length n.

    An ordering of 2n binary sequences with the property that any two consecutive elements differ in exactly one position is called a Gray code.

    Example 5: Gray Code


    Hypercube

    Hypercube

    • Thegraph with one vertex for each n-digit binary sequence and an edge joining vertices that correspond to sequences that differ in just one position is called an n-dimensional cube or hypercube.

    • v = 2n

    • e = n 2n-1


    4 dimensional cube

    4-dimensional Cube.

    0110

    0010

    0111

    1110

    0011

    1010

    1011

    1111

    0001

    1101

    1001

    0000

    0100

    1100

    1000


    4 dimensional cube and a famous painting by salvador dali

    4-dimensional Cube and a Famous Painting by Salvador Dali

    • Salvador Dali (1904 – 1998) produced in 1954 the Crucifixion (Metropolitan Museum of Art, New York) in which the cross is a 3-dimensional net of a 4-dimensional hypercube.


    Gray code revisited

    Gray Code - Revisited

    010

    011

    • A Hamilton path in the hypercube produces a Gray code.

    111

    110

    100

    101

    001

    000


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