UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 5707: Graph theory

Spring 2013


Prerequisites: Math 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.  
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
E-mail: reiner@math.umn.edu 
Classes: Mon-Wed 2:30-4:25pm in Burton Hall 123  
Office hours: Thursdays and Fridays, 2:30-3:20pm. 
Required text: Modern graph theory by B. Bollobas, (1998, Springer Graduate Texts in Mathematics 184).
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 most recent (2012) economics Nobel Prize!),
  • network flows, connectivity,
  • vertex and edge-colorings,
  • perfect graphs
    (subject of a very nice recent 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 to cover (some of) Chapters I, III, V, VI, VII, IX, X in the text by Bollobas, and supplement the text with some outside material.
Other useful texts
Title Author(s), Publ. info Location
Introduction to graph theory
 D. West, Prentice Hall 1996 On reserve in math library
Graph theory R. Diestel The author's download page
Schaum's outlines: graph theory V. K. Balakrishnan On reserve in math library
A course in combinatorial optimization A. Schrijver The author's download page
Homework,
exams,
grading:		 There will be 5 homework assignments due usually every other week, but
  • 2 weeks where there will be a week-long take-home midterm exam,
  • 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.

Late homework will not be accepted. Early homework is fine, and can be left in my mailbox
in the School of Math mailroom near Vincent Hall 105.

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
  • Each of 2 midterms = 20% of grade
  • Final exam = 20% of grade
Tentative HW assignments and exam dates
Assignment or Exam Due date Problems from Bollobas text
Homework 1 Wed, Feb. 6 Chap. I # 1,2,3,7,17,18,19,24,26,35
(Correction in #17, 18, and hint for #19:
In #17, add the assumption that n is at least 2 throughout the problem.
In #18, change "with k components" to "with k components, none of which are isolated vertices".
In #19, as a hint, I might suggest that you characterize the degree sequences of forests that have
a fixed number k of components that are not isolated vertices
along with a fixed number p of isolated vertex components.)
Chap. III # 40, 41
Homework 2 Wed, Feb. 20 Chap. I # 38,85,94
(Typo corrections in #94:
the calligraphic "F" should be a calligraphic "T",
and it should say "diameter at most n-1", not n-2)
Chap. III # 12,18,19,28,82
Exam 1 Wed, Feb. 27 Here is Midterm 1 in PDF.
Homework 3 Wed, Mar. 27 Chap. III # 1, 44, 45, 46, 54, 56
plus these Exercises on orientations in PDF.
Homework 4 Wed, Apr. 10 Chap III # 14
Chap. V # 1,3,5,23,24, 45,46,47,49
(add the hypothesis that G is bridgeless to #23)
Exam 2 Wed, Apr. 17 Here is Midterm 2 in PDF.
Homework 5 Wed, May 1 Chap. VI # 1,2
Chap. VII # 1,2,3,4,6
Chap. X # 1,2,10
Final Exam Wed, May 8 Here is the Final exam in PDF.
A few sources about matching theory in the world around us: Back to Reiner's Homepage.