Math 565: Combinatorics and Graph Theory

Professor: Jonah Blasiak

Fall 2012

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 4-6 pm.
Textbooks: We will mostly follow West. We will use van Lint and Wilson and Stanley for a few topics.
Required text: Introduction to Graph Theory, 2nd Edition, Douglas B. West
Other references:
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. 4, due Oct. 11 in class.
Take home final to be handed out Dec. 11, due 12:30 pm, Dec. 19 (return exams to me in my office).
Grading policy:
  • 30% homework
  • 30% midterm
  • 40% final

    • Tentative Syllabus

    September 4, 6: Graph theory basics: paths, trees, and cycles, Eulerian trails (following parts of Chapters 1 and 2 of West)
    Problem Set 1, Due: Tuesday, September 11
    Note on problem 2: the harder direction requires proving that for any degree sequence of positive integers that sum to 2(n-1), there exists a tree with this degree sequence. (It is not true that any graph with such a degree sequence is a tree. Exercise: find a pair of non-isomorphic simple graphs with the same degree sequence. Find a pair of non-isomorphic connected simple graphs with the same degree sequence.)

    September 11, 13: Kruskal's algorithm, Hall's matching theorem (following parts of Ch. 2 and 3 of West)
    Problem Set 2, Due: Tuesday, September 18
    Note: for problem 5, a matching that covers A is the same as a matching that saturates A.

    September 18, 20: Konig-Egervary theorem, Tutte's 1-factor theorem, connectivity, Menger's theorems (Ch. 3 and 4 of West)
    Problem Set 3, Due: Thursday, September 27

    September 25, 27: Max-flow min-cut theorem, graph coloring (Ch. 4 and 5 of West)
    Problem Set 4, Due: Tuesday, October 2

    October 2, 4: Mycielski's construction, graph minors, planar graphs, Euler's formula (Ch. 5 and 6 of West)
    Midterm, Due: Thursday, October 11
    Note about the midterm: for problem 5, H must be distinct from H'.

    October 9, 11: Haewood's formula, outerplanar graphs, the 5-color theorem, Kuratowski's theorem (Ch. 6 of West)
    Problem Set 5, Due: Tuesday, October 23

    Fall break: No office hours October 15 and no class October 16
    October 18: The perfect graph theorem (following notes by Andras Gyarfas, pages 64-67)

    October 23, 25: 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, October 30

    October 30, November 1: Generating functions, Catalan numbers (Ch. 14 of Van Lint and Wilson)
    Problem Set 7, Due: Tuesday, November 6
    Note: for problem 3, most of these are easy to prove by induction, but the point is to practice generating function manipulations. Define a generating function H for the left hand side and G for the right hand side and show these are the same (for (a)--(c), this can be done by actually computing G and H, but for (d) a more abstract argument is necessary).

    November 6, 8: 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 13

    November 13, 15: 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 20

    November 20: Fisher's inequality and other linear algebra applications (pages 46-56 of notes by Andras Gyarfas)
    No class November 22: Thanksgiving break
    No office hours Wednesday November 21 and Monday November 26.

    November 27, 29: linear algebra applications continued, the chromatic polynomial (Section 5.3 of West)
    Problem Set 10, Due: Tuesday, December 4
    For problem 5, nonnegative eigenvalues must be counted with multiplicities

    December 4, 6: The Tutte polynomial, the bicycle space of graph
    Problem Set 11, Do not turn in (but it may be helpful for the final).

    December 11: Review
    Final, Due: Wednesday, December 19 12:30 pm
    Note: for problem 5, assume that the outerplanar graph is simple.