2.2 Hamilton Circuits

1. 2.2 Hamilton Circuits Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter Hamilton Circuits Hamilton Paths

2. 2.2 Hamilton Circuits a b d c Definition of Hamilton Path: a path that touches every vertex at most once.

3. a b d c 2.2 Hamilton Circuits Definition of Hamilton Circuit: a path that touches every vertex at most once and returns to the starting vertex.

4. a c b d e f g h i k j 2.2 Building Hamilton Circuits Rule 1: If a vertex x has degree 2, both of the edges incident to x must be part of a Hamilton Circuit The red lines indicate the vertices with degree two.

5. a b h i d g f e 2.2 Hamilton Circuit Rule 2: No proper subcircuit, that is, a circuit not containing all vertices, can be formed when building a Hamilton Circuit c

6. a c b d e f g h i k j 2.2 Hamilton Circuit 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. The red lines indicate the edges that have been removed.

7. a c b d e f g h i k j 2.2 Hamilton Circuits Applying the Rules One & Two Rule One: a and g are vertices of degree 2, both of the edges connected to those 2 vertices must be used. Rule Two: You must use all of the vertices to make a Hamilton Circuit, leaving out a vertex would not form a circuit.

8. a c b d e f g h i k j 2.2 Hamilton Circuits Step One: We have two choices leaving i- ij or ik if we choose ij then Rule Three applies. Step Two: Edges jf and ik are not needed in order to have a Hamilton Circuit, so they can be taken out. Step Three: We now have two choices leaving j, jf or jk. If we choose jk, then Rule Three applies and we can delete jf. Applying the Rule Three

9. a b e d c 2.2 Hamilton Circuits Theorem 1 A connected graph with n vertices, n >2, has a Hamilton circuit if the degree of each vertex is at least n/2

10. 2.2 Hamilton Circuits 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

11. 2.2 Hamilton Circuits Theorem 3 Suppose a planar graph G has a Hamilton circuit H. Let G be drawn with any planar depiction, and let ri denote the number of regions inside the Hamilton circuit bounded be i edges in this depiction. 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

12. c b d 4 6 m e a q 6 6 f l p o n 6 4 6 4 g k 6 j h i 2.2 Hamilton Circuit

13. 2.2 Hamilton Circuits Equation in Math Type

14. a c b d 2.2 Hamilton Circuit Theorem 4 Every tournament has a Hamilton path. A tournament is a directed graph obtained from a complete (undirected) graph by giving a direction to each edge. All of the tournaments for this graph are; a-d-c-b, d-c-b-a, c-b-d-a, b-d-a-c, and d-b-a-c.

15. a One answer is; a-g-c-b-f-e-i-k-h-d-j-a f e j g k i h 2.2 Hamilton Circuits Class Work Exercises to Work On (p. 73 #3) Find a Hamilton Circuit or prove that one doesn’t exist. b d c

16. a a-f-b-g-c-h-d-e-a is forced by Rule One, and then forms a subcircuit, violating Rule Two. e f b d i h g c 2.2 Hamilton Circuits Class Work Exercises to Work On Find a Hamilton circuit in the following graph. If one exists. If one doesn’t then explain why.