18.314 | Fall 2014 | Undergraduate
Combinatorial Analysis
Assignments

Problem Set 11

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, 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.)

    • 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+r-1 edges.) Express the chromatic polynomial of K in terms of those of G and H. Example: Three graphs, each made up of rectangles and triangles.* 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.
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Fall 2014
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grading Exams with Solutions
assignment Problem Sets