Problem Session: Mondays 1:00-2:00
Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and Probabilistic Combinatorics. Combinatorics has connections to all areas of mathematics and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a basic knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. The outline for each quarter is the following:
Additional Reading: A graduate level Combinatorics course is surprisingly close to the frontier of research in this area. Approximately, five recent journal articles will be handed out which are related to the current material. Occasionally, problems on the homework will related to this reading.
Exercises: The single most important thing a student can do to learn combinatorics is to work out problems. This is more true in this subject than almost any other area of mathematics. Exercises will be assigned each week but many more good problems are to be found, with solutions, in your text. Do as many of them as you can.
Problem sets: Grading will be based on weekly problem sets due on Wednesdays. The problems will come from the text or from additional reading. Collaboration is encouraged among students. Of course, don't cheat yourself by copying. We will have a problem session on Monday afternoons at 12:45pm to discuss the harder problems.
Grading Problem Sets: Students will be asked to grade another students problem set every other week. More instructions to come with the first assignment.
Computing: Use of computers to verify solutions, produce examples, and prove theorems is highly valuable in this subject. Please turn in documented code if your proof relies on it. If you don't already know a computer language, then try SAGE, Maple or GAP. All three are installed on zeno and theano. Note, theano is our best set of processors for computing.
Monday January 7, 2008 * Lecture 1: Overview, Graph isomorphism invariants.
Wednesday January 9, 2008 * No Lecture:
Friday January 11, 2008 * Lecture 2: Connectivity and trees.
Monday January 14, 2008 * Lecture 3: Spanning trees and the Matrix Tree Theorem
Wednesday January 16, 2008 * Lecture 4: Hamiltonian and Eulerian graphs
Friday January 18, 2008 * Lecture 5: More on Hamiltonian and Eulerian graphs with applications to De Bruijn sequences.
Monday January 21, 2008 * Lecture: No lecture for holiday.
Wednesday January 23, 2008 * Lecture 6: Matchings in bipartite graphs, Konig's thm, Hall's Marriage thm.
Friday January 25, 2008 * Lecture 7: Stable Matchings
Monday January 28, 2008 * Lecture 8: Optimal stable marriages.
Wednesday January 30, 2008 * Lecture 9: Tutte's 1-factor theorem. f-factors.
Friday February 1, 2008 * Lecture 10: Student presentation on Billey-Ardila paper by Gouveia and Pong.
Monday February 4, 2008 * Lecture 11: Characterizing 2,3-connectivity.
Wednesday February 6, 2008 * Lecture 12: Menger's theorem for k-connected graphs.
Friday February 8, 2008 * Lecture 13: Plane graphs. Euler's formula and planar embeddings
Monday February 11, 2008 * Lecture 14: Isomorphism types for plane graphs.
Wednesday February 13, 2008 * Lecture 15: Whitney's thm on embeddings of 3-connected planar graphs. Minors and topological minors of graphs.
Friday February 15, 2008 * Lecture 16: Proof of Kuratowski's theorem.
Monday February 18, 2008 * No Lecture: Holiday
Wednesday February 20, 2008 * Lecture 17: Planarity testing.
Friday February 22, 2008 * Lecture 18: Student presentation on Ciucu-Yan-Zhang paper by Crites and * Merryfield.
Monday February 25, 2008 * Lecture 19: Vertex colorings and chromatic polynomials.
Wednesday February 27, 2008 * Lecture 20: Combinatorial interpretations related to chromatic polynomials. Edge colorings.
Friday February 29, 2008 * Lecture 21: The Four Color Theorem for planar graphs.
Monday March 3, 2008 * Lecture 22: Ramsey Theory
Wednesday March 5, 2008 * Lecture 23: Guest lecture by Carly Klivans "Matrix Tree theorem and Simpicial Tree Theorem"
Friday March 7, 2008 * Lecture 24: Applications of Ramsey Theory.
Monday March 10, 2008 * Lecture 25: Overview of the Graph Minor Theorem.
Wednesday March 12, 2008 * Lecture 26: Overview of matroid theory.
Friday March 14, 2008 * Lecture 27: Crites and Korson present "Forest-Like Permutations" by Bousquet-Melou and Butler.