1 / 16

2.2 Hamilton Circuits

2.2 Hamilton Circuits. Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter. Hamilton Circuits Hamilton Paths. 2.2 Hamilton Circuits. a. b. d. c. Definition of Hamilton Path : a path that touches every vertex at most once. a. b. d. c. 2.2 Hamilton Circuits.

Download Presentation

2.2 Hamilton Circuits

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related