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

3. 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 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,+ if you think you do not need the definition on your cheat sheet, check (if you need just the term as a reminder),check (if you need just the term as a reminder), - if you need more than just the definition to understand the term.- if you need more than just the definition to understand the term.

4. 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.

5. 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.

6. Example 1 Show that the graph on the left has no Hamilton circuit. Hint: Apply Rule 1 four times.

7. Example 2 Show that the graph on the left has no Hamilton circuit. Hint: Apply Rule 1 twice and use symmetry.

8. Example 3 Show that the graph on the left has no Hamilton circuit. Hint: Apply Rule 1 twice and use symmetry. click

9. 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.

10. 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.

11. Theorem 2 (Chvatal, 1972) Let G be a connected graph with n vertices: x 1, x 2,..., x n, so that deg(x 1 ) · deg(x 2 ) ·... · deg(x n ). If for each k · n/2, either deg(x k ) > k or deg(x n-k ) ¸ n – k, then G has a Hamilton circuit.

12. Theorem 3 (Grinberg, 1968) Suppose a plane graph G has a Hamilton circuit H. Let r i 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 r i and r’ i satisfy the equation: (3 - 2)(r 3 – r’ 3 ) + (4 - 2)(r 4 – r’ 4 ) + (5 - 2)(r 5 – r’ 5 ) +... = 0

13. Example 4 Here we have: (a) r 4 + r’ 4 = 3 (b) r 6 + r’ 6 = 6 By Grinberg we should also have: (c) 2(r 4 – r’ 4 ) + 4(r 6 – r’ 6 ) = 0. r 4  r’ 4 ) r 6  r’ 6. Hence |r 6 – r’ 6 | ¸ 2. |r 6 – r’ 6 | ¸ 2 \implies |r 4 – r’ 4 | ¸ 4. Contradiction! The graph in Figure 2.8 has no Hamilton circuit.

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

15. Theorem 4 Every tournament has a (directed) Hamilton path. Proof. By induction on the number of vertices.

16. Example 5: Gray Code Example: n = 3. There are 8 binary sequences: 000 001 010 011 100 101 110 111 There are 2 n binary sequences of length n. An ordering of 2 n binary sequences with the property that any two consecutive elements differ in exactly one position is called a Gray code.

17. Hypercube The graph 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 = 2 n e = n 2 n-1

18. 4-dimensional Cube. 0000 1000 0010 0100 0001 1100 1110 0110 0111 0011 1001 1101 11111011 1010

19. 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.

20. Gray Code - Revisited A Hamilton path in the hypercube produces a Gray code. 000 001 011010 110 100101 111