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 10

- Due in Session 30
- Practice Problems
- Session 25: Chapter 10: Exercise 17
- Session 26: Chapter 11: Exercise 3
- Session 27: Chapter 11: Exercise 3
- Session 28: Chapter 11: Exercises 4, 9, 12, 15, 16

- Problems Assigned in the Textbook
- Do problems 1 and 2 from the Matrix-Tree Theorem (PDF). Problem 2 deserves a star (*). For this problem, you will need the concept of the
*dual*of a planar graph, defined in Definition 12.12, pp. 282, of the text. The first step is to prove that if*G*is a planar graph, then κ(*G*) = κ(*G**). If you are unable to show that κ(*G*) = κ(*G*)*, you should just assume it and continue with the problem. - Chapter 11: Exercise 26. 26(a) is pretty easy, but 26(b) is difficult.

- Do problems 1 and 2 from the Matrix-Tree Theorem (PDF). Problem 2 deserves a star (*). For this problem, you will need the concept of the
- Additional Problems
- (A14*) Let
*G*_{n}denote the complete graph*K*_{2n}with*n*vertex-disjoint edges (i.e., a complete matching) removed. Use the Matrix-Tree theorem to find κ(*G*_{n}), the number of spanning trees of*G*_{n}. (This is not so easy. Find the eigenvalues of the Laplacian matrix*L*of*G*. Several tricks are needed.) - (A15) Let
*m*and*n*be two positive integers. Find the number of Hamiltonian cycles of the complete bipartite graph*K*_{m}_{,n}. (There will be two completely different cases.) - (A16) (a) Let
*G*be the*infinite*graph whose vertices are the points (*i*,*j*) in the plane with integer coordinates, and with an edge between two vertices if the distance between them is one. Is*G*bipartite? (b*) What if the vertices are the same as (a), but now there is an edge between two vertices if the distance between them is an odd integer?

- (A14*) Let
- Bonus Problems
- None