Math 565: Combinatorics and Graph Theory
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.
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)
September 13, 15: Kruskal's algorithm, Hall's matching theorem, Konig-Egervary theorem (following parts of Ch. 2 and 3 of West)
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)
September 27, 29: Max-flow min-cut theorem, graph coloring (Ch. 4 and 5 of West)
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)
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.
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.
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)
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)
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