Math 565: Combinatorics and Graph Theory
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
- A Course in Combinatorics, 2nd Edition, van Lint and Wilson, ISBN
978-0-521-00601-9. Available online here.
- 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. 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).
September 4, 6: Graph theory basics: paths, trees, and cycles, Eulerian trails (following parts of Chapters 1 and 2 of West)
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)
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)
September 25, 27: Max-flow min-cut theorem, graph coloring (Ch. 4 and 5 of West)
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)
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)
on inclusion-exclusion and generalizations. Pages 789-795 are all we need for now. Also see section 3.6 of Stanley.
October 30, November 1: Generating functions, Catalan numbers (Ch. 14 of Van Lint and Wilson)
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.
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)
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)
For problem 5, nonnegative eigenvalues must be counted with multiplicities
December 4, 6: The Tutte polynomial, the bicycle space of graph
December 11: Review
Final, Due: Wednesday, December 19 12:30 pm
Note: for problem 5, assume that the outerplanar graph is simple.