Problem Set 9

Most of the problems are assigned from the required textbook Amazon logo Bona, Miklos. A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory. World Scientific Publishing Company, 2011. ISBN: 9789814335232. [Preview with Google Books]

A problem marked by * is difficult; it is not necessary to solve such a problem to do well in the course.

Problem Set 9

  • Due in Session 25
  • Practice Problems
    • Session 22: Chapter 10: Exercises 1, 2, 7. 7 is quite difficult.
    • Session 23: Chapter 10: Exercises 6, 7, 12, 20
    • Session 24: None
  • Problems Assigned in the Textbook
    • Chapter 10: Exercises 22, 27, 36, 39, 43. In exercise 27, it is not clear what "cross each other" means. What you should prove is that all longest paths have a vertex in common. (This is rather tricky.) For this problem you may assume the result of 26. (As a bonus, you could include a solution to 26). For 43, give a simple combinatorial argument based on Theorem 10.7.
  • Additional Problems
    • (A13*) Give an example of a simple graph with exactly three automorphisms. Note that the graph K3 (a triangle) has six automorphisms.
  • Bonus Problems
    • None