Tuesdays and Thursdays 11:30-1:00, 4151 USB

*A Course in Combinatorics*, 2nd Edition, van Lint and Wilson, ISBN 978-0-521-00601-9. Available online here.*Enumerative combinatorics*, vol.1, R.P.Stanley

Another useful guide, though more geared toward contest problems: How to Write a Solution - by Richard Rusczyk & Mathew Crawford

Take home midterm to be handed out Oct. 4, due Oct. 11 in class.

Take home final to be handed out Dec. 11, due 12:30 pm, Dec. 19 (return exams to me in my office).

September 4, 6: Graph theory basics: paths, trees, and cycles, Eulerian trails (following parts of Chapters 1 and 2 of West)

Problem Set 1, Due: Tuesday, September 11

Note on problem 2: the harder direction requires proving that for any degree sequence of positive integers that sum to 2(n-1),
there exists a tree with this degree sequence. (It is not true that any graph with such a degree sequence is a tree.
Exercise: find a pair of non-isomorphic simple graphs with the same degree sequence.
Find a pair of non-isomorphic connected simple graphs with the same degree sequence.)

September 11, 13: Kruskal's algorithm, Hall's matching theorem (following parts of Ch. 2 and 3 of West)

Problem Set 2, Due: Tuesday, September 18

Note: for problem 5, a matching that covers A is the same as a matching that saturates A.

September 18, 20: Konig-Egervary theorem, Tutte's 1-factor theorem, connectivity, Menger's theorems (Ch. 3 and 4 of West)

Problem Set 3, Due: Thursday, September 27

September 25, 27: Max-flow min-cut theorem, graph coloring (Ch. 4 and 5 of West)

Problem Set 4, Due: Tuesday, October 2

October 2, 4: Mycielski's construction, graph minors, planar graphs, Euler's formula (Ch. 5 and 6 of West)

Midterm, Due: Thursday, October 11

Note about the midterm: for problem 5, H must be distinct from H'.

October 9, 11: Haewood's formula, outerplanar graphs, the 5-color theorem, Kuratowski's theorem (Ch. 6 of West)

Problem Set 5, Due: Tuesday, October 23

Fall break: No office hours October 15 and no class October 16

October 18: The perfect graph theorem (following notes by Andras Gyarfas, pages 64-67)

October 23, 25: Binomial coefficients, inclusion-exclusion, derangements, Euler's phi function (Ch. 10 and 13 of Van Lint and Wilson)

Notes on inclusion-exclusion and generalizations. Pages 789-795 are all we need for now. Also see section 3.6 of Stanley.

Problem Set 6, Due: Tuesday, October 30

October 30, November 1: Generating functions, Catalan numbers (Ch. 14 of Van Lint and Wilson)

Problem Set 7, Due: Tuesday, November 6

Note: for problem 3, most of these are easy to prove by induction, but the point is to practice generating function manipulations. Define a generating function H for the left hand side and G for the right hand side and show these are the same (for (a)--(c), this can be done by actually computing G and H, but for (d) a more abstract argument is necessary).

November 6, 8: More generating functions, eigenvalues of graphs (Section 8.6 of West and Ch. 36 of Van Lint and Wilson)

Spectra of Graphs by Andries Brouwer and Willem Haemers. See page 26 for a table of spectra of small graphs.

Problem Set 8, Due: Tuesday, November 13

November 13, 15: Eigenvalues of graphs, ADE Dynkin diagrams, the matrix tree theorem (Theorem 3.1.3 of Spectra of Graphs, Section 2.2 of West and Ch. 36 of Van Lint and Wilson)

Problem Set 9, Due: Tuesday, November 20

November 20: Fisher's inequality and other linear algebra applications (pages 46-56 of notes by Andras Gyarfas)

No class November 22: Thanksgiving break

No office hours Wednesday November 21 and Monday November 26.

November 27, 29: linear algebra applications continued, the chromatic polynomial (Section 5.3 of West)

Problem Set 10, Due: Tuesday, December 4

For problem 5, nonnegative eigenvalues must be counted with multiplicities

December 4, 6: The Tutte polynomial, the bicycle space of graph

Problem Set 11, Do not turn in (but it may be helpful for the final).

December 11: Review

Final, Due: Wednesday, December 19 12:30 pm

Note: for problem 5, assume that the outerplanar graph is simple.