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](http://books.google.com/books? id=TzJ2L9ZmlQUC&pg=PAfrontcover)]
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, Gxy 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 KonigEgervary theorem.)


Session 31: None

Session 32: Chapter 11: Exercises 6, 7

 Problems Assigned in the Textbook
 Chapter 11: Exercise 30. Hint. Consider the operation ⊕ as used in the proof of Theorem 11.14 on page 260.
 Chapter 11: Exercises 23, 24
 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+r1 edges.) Express the chromatic polynomial of K in terms of those of G and H. Example: * 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.