Most of the problems are assigned from the required textbook 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 _K_
_{3}(a triangle) has*six*automorphisms.

- (A13*) Give an example of a simple graph with exactly three automorphisms. Note that the graph _K_
- Bonus Problems
- None