# Math 4707: Introduction to combinatorics and graph theory

## Fall 2011

 Prerequisites: Math 2243 and either Math 2283 or 3283 (or their equivalent). Students will be expected to know some calculus and linear algebra, as well as having some familiarity with proof techniques, such as mathematical induction. 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 Vincent Hall 207 Office hours: Monday 4-5pm (i.e. right after class), Thursday 12:20-1:10pm, and by appointment. Required text: Discrete Mathematics: elementary and beyond, by Lovasz, Pelikan, and Vesztergombi (2003, Springer-Verlag). Course content: This is a course in discrete mathematics, emphasizing both techniques of enumeration (as in Math 5705) as well as graph theory and optimization (as in Math 5707), but with somewhat less depth than in either of Math 5705 or 5707. We plan to cover most of the above text, skipping Chapters 6, 14, 15. We will also supplement the text with some outside material. Warning: Occasionally some course material will be taught by having the students work together in small groups cooperatively. Students may be asked to come to the board to explain their group's answer. Here are some relevant handouts for these exercises: Poker hand probabilities (by D. Armstrong) Stirling numbers of the 2nd kind
Other useful texts
Title Author(s), Publ. info Location
Invitation to Discrete Mathematics Matousek and Nesetril, Oxford 1998 On reserve in math library
Applied combinatorics A. Tucker, Wiley & Sons 2004 On reserve in math library
Introduction to graph theory D. West, Prentice Hall 1996 On reserve in math library
 Supplementary materials: Derangements (by D. White) Catalan numbers (by D. White) Matchings (by D. White) Chromatic polynomials (by D. White) Generating functions (by G. Musiker) Matrices: walks and spanning trees (by G. Musiker) DeBruijn sequences (by D. Armstrong) Tournament ranking (by D. Armstrong) "The Myth, the Math, the Sex" (by G. Kolata) Ballot theorem proofs (by M. Renault) The Koenigsberg bridges (by J. Barnett) Sloane's Encyclopedia of Integer Sequences 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 Lovasz-Pelikan-Vesztergombi text,
unless otherwise specified
Homework 1 Sept. 28 1.8 # 10,12,14,17,13,21,24,26,27,33
2.5 # 1,3,4(a),5,7,8
3.8 # 4,8,9,11,12
Homework 2 Oct. 12 4.3 # 5,8,9(a,b),11,12
5.4 # 1,2,3,4
Exam 1 Oct. 19 Midterm 1 in PDF
Homework 3 Nov. 2 In this homework, assume graphs are simple,
that is, with no parallel/multiple edges nor self-loops.
In problem 7.3.4, only draw one example of
each such graph, up to isomorphism.
7.3# 4,5,9,10,13
8.5# 2,3,4,5,7,9,11
Homework 4 Nov. 16 9.2# 3, 7
10.4# 5,6,11,13(a,b),15
(10.4.7 was moved to Homework 5)
In 10.4.11, the graph should have 2^3=8 vertices,
and assume they mean a pair of subsets forms an
edge exactly when one is a subset of the other,
and their cardinality differs by one, e.g. {a} and {a,c}.
Exam 2 Nov. 23 Midterm 2 in PDF
Homework 5 Dec. 7 10.4 # 7
12.3 # 1, 2, 5, 6
Correct the hypotheses of 12.3.6 by assuming also
that every vertex has degree 3. And here is a
hint for 12.3.5: first note that the Petersen graph
has no cycles shorter than 5-cycles.
13.4 # 1, 2, 7, 8, 9(a)
Final Exam Dec. 14 Final exam in PDF
Back to Reiner's Homepage.