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 11

- Due in Session 33
- Practice Problems
- Session 30:
- Let
*G*be a bipartite graph with bipartition (*X,Y*). Show that the following three conditions are equivalent.*G*is connected, and each edge of*G*is contained in a perfect matching.- For any
*x*in*X*and*y*in*Y*,*G-x-y*has a perfect matching. *X*=*Y*, and for every nonempty subset*T*of*X*except*X*, we have*N*(*T*) >*T*.

- Let
*M*be an*m*×*n*matrix of 0's and 1's. Let*a*(*M*) be the maximum number of 1's of*M*such that no two are in the same row or column. Let*b*(*M*) be the minimum number of rows and columns of*M*such that cover every 1 (i.e., every one is in at least one of the rows or columns). Show that*a*(*M*) =*b*(*M*). (**Hint.**Use the Konig-Egervary theorem.)

- Let
- Session 31: None
- Session 32: Chapter 11: Exercises

- Session 30:
- Problems Assigned in the Textbook
- Chapter 11: Exercise
**Hint.**Consider the operation ⊕ as used in the proof of Theorem 11.14 on page 260. - Chapter 11: Exercises

- Chapter 11: Exercise
- Additional Problems
- (A17) Let
*G*be a bipartite graph for which a maximum matching has*n*edges. What is the smallest possible size of a*maximal*matching? (You need to give an example of this size and prove that no smaller size is possible in any bipartite graph for which a maximum matching has*n*edges.) - (A18) Let
*G*and*H*be finite graphs. Let*K*consist of the union of*G*and*H*, with an edge*e*of*G*identified with an edge*f*of*H*. (Thus if*G*has*q*edges and*H*has*r*edges, then*K*has*q+r-1*edges.) Express the chromatic polynomial of*K*in terms of those of*G*and*H*. Example:

- (A17) Let
- Bonus Problems
- (B4) Let
*M*be an*n*×*n*matrix of nonnegative integers. What is the least positive integer*f*(*n*) with the following property? If every row and column of*M*sums to*f*(*n*), then there exists*n*entries of*M*, no two in the same row and column, and all greater than one. For instance*f*(2)=3 and*f*(3)=5. The matrix with rows [2,1,1], [2,1,1], [0,2,2] shows that*f*(3)>4.

- (B4) Let