图论与组合 (Fall 2012)

Table of Contents

News

  • 2012/10/15: 期中考试日期10/29，范围是10/17及之前的课的内容。可以带一张A4纸、笔、基本没有计算能力的食物。
  • 2012/10/15: 更新了讲义。

Lecture Notes

本学期的讲义：(ps) - (pdf)

  1. Lecture 1, 2012/09/12 introduction, basic counting, binomial numbers.
  2. Lecture 2, 2012/09/17 Lucas' theorem, combinatorial proofs, incidence algebra on posets.
  3. Lecture 3, 2012/09/19 posets, lattices, Mobius function, and Mobius inversion.
  4. Lecture 4, 2012/09/26 the inclusion-exclusion principle; energy function method.
  5. Lecture 5, 2012/10/10 permanant; classical Mobius inversion on numbers; count the monic irriducible polynomials of degree n; integer patitions.
  6. Lecture 6, 2012/10/15 Euler's pentagonal number theorem; introduction to graphs.

上学年的讲义: (ps) - (pdf)

Homework

  1. Homework1 : (tex) - (ps) - (pdf)
  2. Homework2 : (tex) - (ps) - (pdf)
  3. Homework3 : (tex) - (ps) - (pdf)
  4. Homework4 : Due 2012/10/17 before class (tex) - (ps) - (pdf)

Course Description

Description: This course serves as a broad exploration in the field of combinatorics, with a focus on the topics in or related to the theory of graphs and hyper-graphs. The course starts with the basic enumerative combinatorics, including combinatorial proofs in counting, the inclusion-exclusion principle and Mobius inversion, recursion and generating functions. Then we will discuss many interesting topics and techniques, including Ramsey theorems, extremal graph theory, conbinatorial designs, combinatorial geometry, graph matching, connectivity, planarity, and colouring, random graphs, Szemeredi's regularity lemma, the probabilistic method, and the algebraic method. We will adore the legendary Erdos and his co-authors, and hopefully attack open problems. The course will be self-contained. The students are surely assumed to have the basic ability in problem solving.

Textbook(s) and Articles

教科书：

  • J.H. van Lint and R. M. Wilson: A Course in Combinatorics (2nd ed). Combridge University Press, 2001.

推荐读物：

  • B. Bollobas: Combinatorics. Combridge University Press, 1986.
  • R. Stanley: Enumerative Combinatorics Vol 1, Combridge University Press, 2000.
  • H. Wilf: Generatingfunctionology, A K Peters, 2006. (Also available online)
  • R. Graham, B. Rothschild, and J. Spencer: Ramsey Theory (2nd ed). Wiley-Interscience, 1990.
  • N. Alon and J. Spencer: The probabilistic Method (3rd ed). Wiley-Interscience, 2008.
  • A. Bondy and U.S.R. Murty: Graph Theory with Applications. Elsevier Science Ltd/North-Holland, 1976.
  • A. Bondy and U.S.R. Murty: Graph Theory. Springer 2010.
  • B. Bollobas: Modern Graph Theory. Springer, 1998.
  • D. West: Introduction to Graph Theory (2nd ed). Prentice Hall, 2000.
  • M. Aigner, G. Ziegler, and K. Hofmann: Proofs from the BOOK (4th ed). Springer, 2009.
  • Articles about Paul Erdos: http://www.ams.org/notices/199801/comm-erdos.pdf
  • An interview with Endre Szemeredi: http://www.math.toronto.edu/zsuzsi/research/Szemeredi.pdf

Contact

陈晓敏 gougle [at] gmail [dot] com

金斌 bjin1990+cs477 [at] gmail [got] com

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