UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 5707: Graph theory

Spring 2023



Prerequisites: Linear algebra at the level of Math 2142 or 2243, and either Math 2283 or 3283 (or their equivalent).
Students will be expected to know calculus and linear algebra
(e.g. familiarity with determinants and eigenvalues is expected),
and be ready to read, understand and write proofs.  
Class location, time: Mon-Wed 11:15 AM - 1:10 PM in Akerman Hall 211  
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
E-mail is better: reiner@math.umn.edu 
Office hours: To be determined. 
Discord server: Here is our class Discord server.
Using it to discuss homework and form study groups are strongly encouraged. 
Required text: This is our only required text, a free PDF:
Graph theory with applications by J.A. Bondy and U.S.R. Murty
Course content: 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, along with some other supplementary material.
Other useful texts
Title Author(s), Publ. info
Introduction to graph theory
 D. West, Prentice Hall 1996
Modern Graph Theory
 B. Bollobas, Springer Graduate Texts in Math
Graph theory R. Diestel, The author's download page
Schaum's outlines: graph theory V. K. Balakrishnan
A course in combinatorial optimization A. Schrijver, The author's download page
Homework,
exams,
grading:		 There will be 5 homework assignments due usually every 2-3 weeks, but
  • there are 2 weeks with a week-long take-home midterm exam,
  • and a week at the end with a week-long take-home final exam.
Tentative dates for the assignments and exams are in the schedule below.

I encourage collaboration on the homework, as long as each person understands the solutions,
writes them up in their own words, and indicates on the homework page their collaborators.
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.

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.

Late homework will not be accepted.

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.
Grading scheme :
  • Homework = 40% of grade
  • Midterm 1 = 20% of grade
  • Midterm 2 = 20% of grade
  • Final exam = 20% of grade
Tentative HW assignments and exam dates
Assignment,
Exam
, other event 		Due date Problems
(from Bondy and Murty,
unless specified otherwise)
Homework 1 Wed, Feb 8 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 22 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 1 Here is Midterm 1 in PDF.
No in-class lectures
Mon. Mar 13,
Wed. Mar 15		 Hall's theorem (video) and two applications (video), with notes for both
Weighted bipartite matching (video), with notes
Nonbipartite matching: Edmonds' "blossom" algorithm (video), with notes
Homework 3 Wed, Mar 22 Section 5.1: #1,2,5(a)(i,ii)
Section 5.2: #1,4,5
(removed Section 5.5: #1)
Homework 4 Wed, Apr 5 Section 11.1: #1
Section 11.2: #1,2,3
Section 3.1: #1,2
Section 3.2: #2
No in-class lectures
Mon. Aor 3,
Wed. Apr 5		 Vertex-coloring (video), with notes
Brooks's Theorem (video), with notes
Edge-coloring (video), with notes
Midterm exam 2 Wed, Apr 12 Here is Midterm 2 in PDF.
Homework 5 Wed, Apr 26 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 3 Here is the Final exam in PDF.

A few sources about matching theory in the world around us:
A few more resources
Topic Author Title/info
List of open problems and
conjectures in graph theory 		Doug West's
Bonato and Nowakowski's
IRMACS 		problems page
Sketchy Tweets: 10 minute conjectures in graph theory
Open Problem Garden for Graph Theory
Probabilistic method N. Alon and J. Spencer The probabilistic method
Wiley-Interscience, 2000
Surfaces and graphs on them W.S. Massey (Chap. 1 of) Algebraic topology: an introduction
Springer-Verlag Graduate Texts in Math 56
P. Giblin Graphs, surfaces and homology
Cambridge Univ. Press 2010
Back to Reiner's Homepage.