Math 5707: Graph theory

Spring semester 2025

Mon-Wed 11:15 AM - 1:10 PM in Amundson Hall 162
Course Canvas page, and Discord server.

Course Instructor

Victor Reiner (You can call me "Vic").
Office: Vincent Hall 256
Telephone (with voice mail): (612) 625-6682
E-mail is better: reiner@math.umn.edu
Office hours: Mon and Tues from 4:40-5:30pm
in my office Vincent 256, and on this Zoom link.

Course Schedule

HW assignments and exam dates
Assignment,
Exam
or other event Due date Problems
(from Bondy and Murty,
unless specified otherwise)

Homework 1 Wed, Feb 12 Section 1.2: #4,10,11
Section 1.4: #4
Section 1.5: 5,7(a),10
Section 1.6: #7,10
Section 1.7: #2
Section 4.1: #1,2
Section 4.2: #2,3

Homework 2 Wed, Feb 26 Section 2.1: #6,12
Section 2.2: #2,3,5
Section 2.4: #1,5
Section 2.5: #3,5

Midterm exam 1 Wed, Mar 5 Here is Midterm 1 in PDF.

Spring Break, March 10-14

Homework 3 Wed, Mar 26 Section 5.1: #1,2,5(a)(i,ii)
Section 5.2: #1,4,5

Homework 4 Wed, Apr 9 Section 11.1: #1
Section 11.2: #1,2,3
Section 3.1: #1,2
Section 3.2: #2

Midterm exam 2 Wed, Apr 16 Here is Midterm 2 in PDF.

Homework 5 Wed, Apr 30 Section 8.1: #3,4
Section 8.4: #3,4
Section 9.2: #2
Section 9.3: #3(a)
Section 9.6: #2 (actually only show half of this one:
show that if a plane triangulation is 3-colorable, then it must be eulerian.)

Final exam Wed, May 7 Final exam will appear here in PDF.

Lecture notes
Date(s) discussed Lecture topics
(with section from Garrett text indicated)

Wed Jan 22, Mon Jan 27 Intro and Degree Sequences (Sec 1.1,1.2,1.5)

Wed Jan 29, Mon Feb 3, Wed Feb 5 Euler and Hamilton Cycles (Sec 4.1,4.2)

Mon Feb 10, Wed Feb 12 Complexity and P vs NP vs NP-complete (not in book)

Wed Feb 12, Mon Feb 17 Trees (Chapter 2)

Mon Feb 17, Wed Feb 19, Mon Feb 24, Wed Feb 26 Counting trees (Sec. 2.4) and directed Euler tours (not in book)

Mon Mar 3, Wed Mar 5, Mon Mar 17, Wed Mar 19 Matching theory (Chapter 5)

Mon Mar 24, Wed Mar 26, Mon Mar 31 Network flows (Chap. 11), Menger's Theorems, connectvity (Chap. 3)

Wed Apr 2, Mon Apr 7, Wed Apr 9 Vertex and edge-colorings (Chaps. 8, 6) and perfect graphs (not in book),
Proof of Weak Perfect Graph Conjecture (video).

Mon Apr 14, Wed Apr 16, Mon Apr 21 Planarity (Chap. 9) and graph minors

Mon Apr 21, Wed Apr 23 Chromatic polynomials (Sec. 8.4)

Wed Apr 23, Mon Apr 28, Wed Apr 30 Ramsey Theory (Sec. 7.2) and Probabilistic Method (not in book)

Wed Apr 30, Mon May 5 More on planar duality

Mon May 5, Wed May 7 Coloring graphs on surfaces

**HW assignments and exam dates**
Assignment, Exam or other event	Due date	Problems (from Bondy and Murty, unless specified otherwise)
Homework 1	Wed, Feb 12	Section 1.2: #4,10,11 Section 1.4: #4 Section 1.5: 5,7(a),10 Section 1.6: #7,10 Section 1.7: #2 Section 4.1: #1,2 Section 4.2: #2,3
Homework 2	Wed, Feb 26	Section 2.1: #6,12 Section 2.2: #2,3,5 Section 2.4: #1,5 Section 2.5: #3,5
Midterm exam 1	Wed, Mar 5	Here is Midterm 1 in PDF.
Spring Break, March 10-14
Homework 3	Wed, Mar 26	Section 5.1: #1,2,5(a)(i,ii) Section 5.2: #1,4,5
Homework 4	Wed, Apr 9	Section 11.1: #1 Section 11.2: #1,2,3 Section 3.1: #1,2 Section 3.2: #2
Midterm exam 2	Wed, Apr 16	Here is Midterm 2 in PDF.
Homework 5	Wed, Apr 30	Section 8.1: #3,4 Section 8.4: #3,4 Section 9.2: #2 Section 9.3: #3(a) Section 9.6: #2 (actually only show half of this one: show that if a plane triangulation is 3-colorable, then it must be eulerian.)
Final exam	Wed, May 7	Final exam will appear here in PDF.

**Lecture notes**
Date(s) discussed	Lecture topics (with section from Garrett text indicated)
Wed Jan 22, Mon Jan 27	Intro and Degree Sequences (Sec 1.1,1.2,1.5)
Wed Jan 29, Mon Feb 3, Wed Feb 5	Euler and Hamilton Cycles (Sec 4.1,4.2)
Mon Feb 10, Wed Feb 12	Complexity and P vs NP vs NP-complete (not in book)
Wed Feb 12, Mon Feb 17	Trees (Chapter 2)
Mon Feb 17, Wed Feb 19, Mon Feb 24, Wed Feb 26	Counting trees (Sec. 2.4) and directed Euler tours (not in book)
Mon Mar 3, Wed Mar 5, Mon Mar 17, Wed Mar 19	Matching theory (Chapter 5)
Mon Mar 24, Wed Mar 26, Mon Mar 31	Network flows (Chap. 11), Menger's Theorems, connectvity (Chap. 3)
Wed Apr 2, Mon Apr 7, Wed Apr 9	Vertex and edge-colorings (Chaps. 8, 6) and perfect graphs (not in book), Proof of Weak Perfect Graph Conjecture (video).
Mon Apr 14, Wed Apr 16, Mon Apr 21	Planarity (Chap. 9) and graph minors
Mon Apr 21, Wed Apr 23	Chromatic polynomials (Sec. 8.4)
Wed Apr 23, Mon Apr 28, Wed Apr 30	Ramsey Theory (Sec. 7.2) and Probabilistic Method (not in book)
Wed Apr 30, Mon May 5	More on planar duality
Mon May 5, Wed May 7	Coloring graphs on surfaces

Course Summary

Graphs are networks of vertices (nodes) connected by edges.
They are interesting objects in mathematics, but also usefully model
problems in computer science, optimization, and social science.
This is a first course in graph theory, emphasizing classical topics, such as

vertex degrees,
Euler and Hamilton circuits,
trees and Laplacian matrices,
matching theory and stable matchings (subject of the 2012 economics Nobel Prize),
network flows, connectivity,
vertex and edge-colorings,
perfect graphs (subject of a nice survey),
planarity and graphs on surfaces,
duality (a class exercise handout)
deletion-contraction, Tutte polynomials,
(a tiny bit of) probabilistic methods.

This course contrasts with some related courses in our curriculum,
in that the material is

covered at a more sophisticated and rigorous level than Math 4707,
focused less on enumerative questions than Math 5705, and
focused less on optimization than Math 5711/IE 5531

We plan on covering some of Chapters 1-9 and 11 of the Bondy-Murty text linked below,
along with some other supplementary material.

Course Prerequisites

Linear algebra at the level of Math 2142 or 2243 or 4242,
and familiarity with proofs as in either Math 2283 or 3283 (or their equivalent).
Students will be expected to know calculus and linear algebra
(for example, familiarity with determinants and eigenvalues is expected),
and be ready to read, understand and write proofs.

Course Goals and Objectives

By the end of this course, here are a few things that we hope a student will have learned:

How complexity theory lets one quantify the difficulty of some problems compared to others.
Which of the problems in graph theory appear to be easier versus harder in this complexity sense,
including some easy versus hard problems that sound very similar.
How ideas from linear algebra, probability, topology play a role in the graph theory.
How graphs and networks can model problems that arise in the world, and graph algorithms can be used to solve them.
Some of the history of the Four Color Problem, and philosophical issues raised by its computational proof.

Course Format

We mainly use a lecture format during the Monday and Wednesday class periods,
interspersed with occasional active learning exercises.

Course links

Course Canvas page.
Our free text: Graph theory with applications by Bondy and Murty
Here is the class Discord server. Using it to discuss homework and form study groups are strongly encouraged.
Other useful texts
- Introduction to graph theory, by D. West, Prentice Hall 1996
- Modern Graph Theory by B. Bollobas, Springer Graduate Texts in Math
- Graph theory by R. Diestel, The author's download page
- An introduction to graph theory by D. Grinberg, course notes
- Graph theory by B. Sudakov, course notes
- Schaum's outlines: graph theory, by V. K. Balakrishnan
- A course in combinatorial optimization, by A. Schrijver, a free text
A few more resources:
- Lists of open problems and conjectures in graph theory:
  - IRMACS Open Problem Garden for Graph Theory
- A classic text on The probabilistic method by N. Alon and J. Spencer, Wiley-Interscience, 2000
- For background on graphs on surfaces
  - (Chap. 1 of) Algebraic topology: an introduction by W.S. Massey, Springer-Verlag Graduate Texts in Math 56
  - Graphs, Surfaces and Homology by P.J. Giblin, Cambridge Univ. Press

Textbook and Required Materials

The only required text is this free PDF, also linked above: Graph theory with applications by J.A. Bondy and U.S.R. Murty

Course Grading

The course grade is based on a combination of homework and exams.
There are 5 homeworks due every two weeks on Wednesday evenings around midnight,
two take-home midterm exams and a take-home final exam. All homework and exams are to be turned in at the course Canvas site.
See the schedule table above for the assignments and dates.
All three exams are in a weeklong take-home format.

Exams

Midterm exam 1 due Wed, March 5
Midterm exam 2 due Wed, April 16
Final exam due Wed, May 7

Grading

Homework = 40% of grade
Midterm 1 = 20% of grade
Midterm 2 = 20% of grade
Final exam = 20% of grade

Course Policies

I encourage collaboration on the homework as long as each person understands the solutions, writes them up in their own words.
Indicate on the homework page any collaborators, and/or resources used, including AI tools such as ChatGPT.
Homework solutions should be well-explained-- the grader is told not to give credit for an unsupported answer.
Complaints about the grading should be brought to me.
The take-home midterms and final exam are open-book, open-library, open-web,
but in contrast to the homework on exams, no collaboration or consultation of human sources is allowed.
Again, on exams indicate any resources used, including AI tools such as ChatGPT.

We will only be using the course Canvas site for turning in the homeworks and exams as PDFs.
If you write solutions by hand, then use a scanning app (e.g., Adobe Photo Scan) or a scanner to create the PDFs.
Do not just take a photo and convert it to PDF, as those are harder to read.

Attendance

I will not be checking attendance, but attending all lectures is strongly encouraged.
There is a significant amount of material that will appear in lectures,
which is not in the Bondy and Murty text, but gleaned from some of the other sources in this syllabus.

Make-up/Dropped Exams and Late Work

Late homework will not be accepted, without accommodations from Disability Servcies
or medical excuses written by a medical professional.
There will be no make-up or dropped exams or homeworks.

Incompletes

The grade I ("incomplete") shall be assigned at the lecturer's discretion when, due to extraordinary circumstances,
the student was prevented from completing the entire course. It is my policy to assign incompletes only rarely,
and only when almost all of the course has already been completed in a satisfactory fashion prior to the extraordinary circumstances.
See me if something occurs which makes you think you should receive an incomplete.

Required University Policy Statements

Education & Student Life Policies
Student Conduct Code
Use of Personal Electronic Devices in the Classroom
Scholastic Dishonesty
Excused Absences and Makeup Work
Appropriate Student Use of Class Notes and Course Materials
University Grading Scales
Sexual Harassment, Sexual Assault, Stalking and Relationship Violence
Equity, Diversity, Equal Opportunity, and Affirmative Action
Disability Accommodations
Mental Health and Stress Management