Math 1230, graph theory
This is a theoretical and proofs-based introduction to graph theory.
We will cover classic results such as spanning trees, network flow problems, matching problems, coloring problems, planarity, Cayley graphs, Ramsey Theory, etc.
You'll see a combination of theory and algorithms, with preference for the theory.
Techniques from graph theory are widely applicable in many other areas of mathematics, including geometry, combinatorics, and algebra. This couse will aim to highlight some of these connections.
A few key points from the syllabus:
- Prerequisites: Linear algebra (we will use it!) and ability to write proofs and problem-solve independently.
- My office hours: Wednesdays 1:45 - 2:45, Thursdays 10:30 - 11:30.
- Tutorial/problem session: Moved to: Watson Institute, 111 Thayer, room 116. Monday 4:30-6:30
- Lectures: Tu/Th 9:00 - 10:20 am in Barus and Holley (BH) 153. You are required to attend, arrive on time, and participate!
- Textbook: Introduction to Graph Theory, second edition by Douglas B. West.
- See the syllabus for homework policy, grade scheme, credit hours, etc.
Outside the classroom:
- Interested in indepenent study or learning your own thing? Check out the Directed Reading Program a grad-undergrad math reading mentorship program.
- The math department wants to hear what you have to say! There's a anonymous suggestion box and info on inclusivity on this page.
- The Brown math DUG is another great resource for all kinds of info.
Problem sets and week-by-week reading
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1/29 and 1/31. Reading: West, sections 1.1 and 1.2.
Homework due 2/5.
Just for fun: an article on Euler and the Konigsberg bridge problem
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2/5 and 2/7. Reading: West, sections 1.3 (we will not cover all of it)
Homework due 2/12. Note: question 4 should assume the graph is connected
Additional reading/references for this week:
1. Sperner's lemma is discussed and proved on West, pages 388-390 (we will not cover the applicaiton to bandwith, but you might find it interesting)
2. Reading on Sperner's theorem and odd numbers of triangles, covered in Thursday's lecture. From "Proofs from the book" by Aigner and Ziegler.
- 2/12 and 2/14. Reading: West, section 1.4
Homework due 2/21 (no class 2/19). Don't forget to study for the upcoming quiz!
Additional reading/references for this week:
Nim handout from Feb. 12
Why are de Bruijn graphs useful for genome assembly?
- 2/21. 30-minute quiz + Trees and spanning trees. Reading: West, section 2.1
Short homework due 2/26.
- 2/26 and 2/28. Reading: West, section 2.1, some of 2.2 (we will not cover all the subsections)
Homework due 3/5.
Extra (optional) reading if you're interested:
Quiz Solutions will be posted Thursday when quizzes are returned.
- 3/5 and 3/7. Linear algebra and graph theory; intro to matchings.
Reading: the Matrix tree Theorem in West 2.2, Section 3.1.
You might also find the first sub-section of 8.6 helpful for some of the linear algebra (just the part with heading "the characteristic polynomial")
Homework due 3/12.
Supplimental reference: Proof of Cauchy-Binet (notes of Prof. Schwartz) or wikipedia version
- 3/12 and 3/14.
On 3/12 we did more matching problems
There is a midterm on 3/14. For the midterm, you are responsible for everything up to section 3.1. (the content of the last week.)
SOLUTIONS to the midterm
- 3/19 and 3/21. Latin squares, weighted matchings.
Homework due 4/2.
Required viewing: Hungarian algorithm step-by-step and worked example. (By D. Stolee, former instructor at UIUC, now researcher at Microsoft)
Required reading: Latin squares in experiment design
Optional reading (just for fun): 1. Kuhn's original paper on the Hungarian algortihm (look at the author's introduction!)
2. Ryser's original paper on Latin squares
- 4/2 and 4/4. Stable matchings, the Dinitz problem, max-flow/min-cut.
Reading: West, p. 130, the Dinitz Problem , West 4.3.
Homework due 4/9.
- 4/9 and 4/11. Coloring problems, Brook's theorem.
Reading: West 5.1 and the first part of 5.3.
Homework due 4/16.
- 4/16 and 4/18. Ramsey theory, the probabilistic method.
Reading: West 8.3 sections on Ramsey Theory and Ramsey Numbers; the very beginning of 8.5
Homework due 4/23.
Optional reading on random graphs, if you are interested in learning more: Diestel's book chapter available online here, West section 8.5.
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4/23 and 4/25. The Rado graph, "THE" random graph, graphs with infinitely many vertices and edges.
Reference: notes written by Prof. Schwartz
Last homework (short) due 4/30.
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Week of 4/29 No class (reading period).
HW11 is due in my mailbox by 5:00pm on Tues, April 30.
Reading assignment: Extremal and Probabilistic Combinatorics from the Princeton Companion to Mathematics.
This is a panorama of Ramsey theory, probabilistic methods, matching and coloring problems, and more.
I would like you to read it carefully, and e-mail me a short response as indictated on HW11.
Extra office hours and a final review session before the exam will be posted here and announced by e-mail.
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Week of 5/6
Review session: Monday 4:30-6:30 pm in 111 Thayer room 116.
Office hours: Tuesday 4:30pm - 6pm, Wednesday 11:30am - 1pm in my office.
Final exam: Fri, May 10, 9:00 am in Barus & Holley 168
Here is some addional materal to help you review for the final