Math 565: Combinatorics and Graph Theory

Professor: Jonah Blasiak

Fall 2011

Tuesdays and Thursdays 11:30-1:00, 4151 USB

Course Description: combinatorics is the study of counting, organizing, and optimizing over discrete objects. We will introduce some of the basic objects and methods in combinatorics, with an emphasis on graph theory. Problem solving will be emphasized.
Prerequisites: linear algebra and some exposure to proofs and abstract mathematics.
Level: mixed undergraduate and graduate.
Office Hours: 3831 East Hall, Wednesday, 4 - 5:30 pm.
Office Hours/Problem Session: 2nd floor commons - East Hall, Monday 6-7, 8-9 pm.
Textbooks: van Lint and Wilson is a good reference and contains almost all of the material we will cover, however I prefer the treatment in West for most of the graph theory topics. I will try to make at least some parts of West available here or on ctools.
Required text: A Course in Combinatorics, 2nd Edition, van Lint and Wilson, ISBN 978-0-521-00601-9. Available online here.
Recommended text: Introduction to Graph Theory, 2nd Edition, Douglas B. West
Other references:
  • Enumerative combinatorics, vol.1, R.P.Stanley
  • Homework Policy: You may consult each other, the library, the internet and any other source for aid provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. It is likely that you will be able to find solutions to some of the problems if you look hard enough. Being able to search and read through literature is a useful skill, but is not the main focus of this class. It is recommended that you reserve extensive literature searches for only the hardest problems.
    Guidelines for problem set writeups
    Another useful guide, though more geared toward contest problems: How to Write a Solution - by Richard Rusczyk & Mathew Crawford
    Take home exam policy: You may not consult with other people or outside sources; you may consult your notes and the textbooks West, van Lint and Wilson, and Stanley above, as well as any handouts I provide.
    Take home midterm to be handed out Oct. 6, due Oct. 13 in class.
    Take home final handed out in class on December 13, due at 3:30 pm December 20 (return exams to me in my office).
    Grading policy:
  • 30% homework
  • 30% midterm
  • 40% final

  • Tentative Syllabus

    September 6, 8: Graph theory basics: paths, trees, and cycles, Eulerian trails (following parts of Chapters 1 and 2 of West)
    Problem Set 1, Due: Tuesday, September 13

    September 13, 15: Kruskal's algorithm, Hall's matching theorem, Konig-Egervary theorem (following parts of Ch. 2 and 3 of West)
    Problem Set 2, Due: Tuesday, September 20
    Note: for problem 5, a matching that covers A is the same as a matching that saturates A.

    September 20, 22: Tutte's 1-factor theorem, connectivity, Menger's theorems (Ch. 3 and 4 of West)
    Problem Set 3, Due: Tuesday, September 27

    September 27, 29: Max-flow min-cut theorem, graph coloring (Ch. 4 and 5 of West)
    Problem Set 4, Due: Tuesday, October 4

    October 4, 6: Mycielski's construction, graph minors, planar graphs, Euler's formula (Ch. 5 and 6 of West)
    Midterm, Due: Thursday, October 13

    October 11, 13: Haewood's formula, outerplanar graphs, Kuratowski's theorem (Ch. 6 of West)
    Problem Set 5, Due: Tuesday, October 25

    No class October 18: Fall break
    October 20: The perfect graph theorem (following notes by Andras Gyarfas, pages 64-67)

    October 25, 27: Binomial coefficients, inclusion-exclusion, derangements, Euler's phi function (Ch. 10 and 13 of Van Lint and Wilson)
    Notes on inclusion-exclusion and generalizations. Pages 789-795 are all we need for now. Also see section 3.6 of Stanley.
    Problem Set 6, Due: Tuesday, November 1

    November 1, 3: Generating functions, Catalan numbers (Ch. 14 of Van Lint and Wilson)
    Problem Set 7 (Problem 1 has been corrected), Due: Tuesday, November 8

    November 8, 10: More generating functions, eigenvalues of graphs (Section 8.6 of West and Ch. 36 of Van Lint and Wilson)
    Spectra of Graphs by Andries Brouwer and Willem Haemers. See page 26 for a table of spectra of small graphs.
    Problem Set 8, Due: Tuesday, November 15

    November 15, 17: Eigenvalues of graphs, ADE Dynkin diagrams, the matrix tree theorem (Theorem 3.1.3 of Spectra of Graphs, Section 2.2 of West and Ch. 36 of Van Lint and Wilson)
    Problem Set 9, Due: Tuesday, November 22

    November 22: Fisher's inequality and other linear algebra applications (pages 46-56 of notes by Andras Gyarfas)
    No class November 24: Thanksgiving break

    November 29, December 1: linear algebra applications continued, the chromatic polynomial (Section 5.3 of West)
    Problem Set 10, Due: Tuesday, December 6
    For problem 5, nonnegative eigenvalues must be counted with multiplicities

    December 6, 8: The Tutte polynomial, Knot invariants
    Problem Set 11, Do not turn in (but it may be helpful for the final). Problem 5 was wrong; I think it is now correct.

    December 13: Review
    Final, Due: Tuesday, December 20
    Office hours this week: Wednesday 4-5:30, Friday 12-1, Monday 12-1