Readings

The following textbooks are the main textbooks for the class:

Stanley, R. P. Enumerative Combinatorics. Vol. I and II. Cambridge, UK: Cambridge University Press, 1999. ISBN: 0521553091 (hardback: vol. I); 0521663512 (paperback: vol. I); 0521560691 (hardback: vol. II).

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998. ISBN: 0387984917.

———. Extremal Graph Theory. New York, NY: Dover, 2004. ISBN: 0486435962.

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000. ISBN: 3540663134.

The following textbooks can be used as supplemental reading:

Diestel, R. Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1997. ISBN: 3540261834. (Available electronically on the Graph Theory Web site by R. Diestel).

Matousek, J. Lectures on Discrete Geometry (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 2002. ISBN: 0387953736.

The following readings specifically deal with problem 6 from Problem Set 1:

The original paper is here:

Burago, Ju. D., and V. A. Zalgaller. “Polyhedral embedding of a net.” Vestnik Leningrad Univ 15 (1960): 66-80. (In Russian)

A recent relatively simple solution:

Maehara, H. “Acute triangulations of polygons.” European J Combin 23 (2002): 45-55.

Interestingly enough, if one allows right triangles there exist plentiful literature:

Baker, B. S., E. Grosse, and C. S. Rafferty. “Nonobtuse triangulation of polygons.” Discrete Comput Geom 3 (1988): 147-168.

Bern, M., and D. Eppstein. “Polynomial-size nonobtuse triangulation of polygons.” Internat J Comput Geom Appl 2 (1992): 241-255; Errata 449-450.

Bern, M., S. Mitchell, and J. Ruppert. “Linear-size nonobtuse triangulation of polygons.” Discrete Comput Geom 14 (1995): 411-428.

The following table lists the readings assigned for each lecture.

Lec # Topics Readings
1 Course Introduction

Ramsey Theorem

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 182-189. ISBN: 0387984917.
2 Additive Number Theory

Theorems of Schur and Van der Waerden

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 326. ISBN: 3540663134.

Khinchin, A. Y. Three Pearls of Number Theory. Mineola, NY: Dover Publications, Inc., 1998, section 1. ISBN: 0486400263. (Reprint of the 1952 translation.)

3 Lower Bound in Schur’s Theorem

Erdös-Szekeres Theorem (Two Proofs)

2-Colorability of Multigraphs

Intersection Conditions

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 230, 327, and 65-66. ISBN: 3540663134.
4 More on Colorings

Greedy Algorithm

Height Functions Argument for 3-Colorings of a Rectangle

Erdös Theorem

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 66-67. ISBN: 3540663134.

Luby, M., D. Randall, and A. Sinclair. “Markov Chain Algorithms for Planar Lattice Structures.” FOCS 1995. (Paper)

5 More on Colorings (cont.)

Erdös-Lovász Theorem

Brooks Theorem

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 67. ISBN: 3540663134.

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 145-149. ISBN: 0387984917.

6 5-Color Theorem

Vizing’s Theorem

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 146-154. ISBN: 0387984917.

———. Extremal Graph Theory. New York, NY: Dover, 2004, pp. 221-234. ISBN: 0486435962.

7 Edge Coloring of Bipartite Graphs

Heawood Formula

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 154-161. ISBN: 0387984917.

———. Extremal Graph Theory. New York, NY: Dover, 2004, pp. 243-254. ISBN: 0486435962.

8 Glauber Dynamics

The Diameter

Explicit Calculations

Bounds on Chromatic Number via the Number of Edges, and via the Independence Number

 
9 Chromatic Polynomial

NBC Theorem

 
10 Acyclic Orientations

Stanley’s Theorem

Two Definitions of the Tutte Polynomial

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 335-339. ISBN: 0387984917.
11 More on Tutte Polynomial

Special Values

External and Internal Activities

Tutte’s Theorem

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 345-354. ISBN: 0387984917.
12 Tutte Polynomial for a Cycle

Gessel’s Formula for Tutte Polynomial of a Complete Graph

Gessel, I. M. “Enumerative applications of a decomposition for graphs and digraphs.” Discrete Math 139, no. 1-3 (1995): 257–271. (Paper)
13 Crapo’s Bijection

Medial Graph and Two Type of Cuts

Introduction to Knot Theory

Reidemeister Moves

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 358-363. ISBN: 0387984917.

Korn, M., and I. Pak. Combinatorial evaluations of the Tutte polynomial. Preprint (2003) available at Research (Igor Pak Home Page). (Paper)

14 Kauffman Bracket and Jones Polynomial Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 364-371. ISBN: 0387984917.
15 Linear Algebra Methods

Oddtown Theorem

Fisher’s Inequality

2-Distance Sets

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, section 14. ISBN: 3540663134.
16 Non-uniform Ray-Chaudhuri-Wilson Theorem

Frankl-Wilson Theorem

 
17 Borsuk Conjecture

Kahn-Kalai Theorem

Aigner, M., and G. Ziegler. Proof from the BOOK. 2nd ed. New York, NY: Springer-Verlag, August 1998, pp. 83-88. ISBN: 3540636986.
18 Packing with Bipartite Graphs

Testing Matrix Multiplication

 
19 Hamiltonicity, Basic Results

Tutte’s Counter Example

Length of the Longest Path in a Planar Graph

Diestel, R. Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1997, section 10.1. ISBN 3540261834. (Available electronically on the Graph Theory Web site by R. Diestel).
20 Grinberg’s Formula

Lovász and Babai Conjectures for Vertex-transitive Graphs

Dirac’s Theorem

Bollobás, B. Extremal Graph Theory. New York, NY: Dover, 2004, pp. 143-146. ISBN: 0486435962.
21 Tutte’s Theorem

Every Cubic Graph Contains either no HC, or At Least Three

Examples of Hamiltonian Cycles in Cayley Graphs of Sn

 
22 Hamiltonian Cayley Graphs of General Groups Pak, I., and R. Radoicic. “Hamiltonian paths in Cayley graphs.” Preprint (2002) available at Research (Igor Pak Home Page). (Paper)
23 Menger Theorem

Gallai-Milgram Theorem

Diestel, R. Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1997, sections 2.5, and 3.3. ISBN 3540261834. (Available electronically on the Graph Theory Web site by R. Diestel).
24 Dilworth Theorem

Hall’s Marriage Theorem

Erdös-Szekeres Theorem

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 38-39, and 97-100. ISBN: 3540663134.
25 Sperner Theorem

Two Proofs of Mantel Theorem

Graham-Kleitman Theorem

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 40-41, and 45-46. ISBN: 3540663134.
26 Swell Colorings

Ward-Szabo Theorem

Affine Planes

Jukna, S. Extremal Combinatorics. New York, NY: Springer-Verlag, Berlin, 2000, pp. 43-45, and 161-163. ISBN: 3540663134.
27 Turán’s Theorem

Asymptotic Analogues

Bollobás, B. Modern Graph Theory (Graduate Texts in Mathematics). New York, NY: Springer-Verlag, 1998, pp. 108-111. ISBN: 0387984917.
28 Pattern Avoidance

The case of S3 and Catalan Numbers

Stanley-Wilf Conjecture

 
29 Permutation Patterns

Arratia Theorem

Furedi-Hajnal Conjecture

Arratia, R. “On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern.” Electron J Combin 6, no. 1 (1999). (Paper)
30 Proof by Marcus and Tardos of the Stanley-Wilf Conjecture Marcus, A., and G. Tardos. “Excluded permutation matrices and the Stanley-Wilf conjecture.” J Combin Theory Ser A 107, no. 1 (2004): 153–160.
31 Non-intersecting Path Principle

Gessel-Viennot Determinants

Binet-Cauchy Identity

Stanley, R. P. Enumerative Combinatorics. Vol. I. Cambridge, UK: Cambridge University Press, 1999, section 2.7. ISBN: 0521553091 (hardback : vol. I); 0521663512. (paperback : vol. I).
32 Convex Polyomino

Narayana Numbers

MacMahon Formula

Stanley, R. P. Enumerative Combinatorics. Vol. II. Cambridge, UK: Cambridge University Press, 1999, pp. 378. ISBN: 0521560691 (hardback: vol. II).
33 Solid Partitions

MacMahon’s Theorem

Hook-content Formula

Stanley, R. P. Enumerative Combinatorics. Vol. II. Cambridge, UK: Cambridge University Press, 1999, section 7. ISBN: 0521560691 (hardback: vol. II).
34 Hook Length Formula Pak, I. “Hook Length Formula and Geometric Combinatorics.” Séminaire Lotharingien de Combinatoire 46 (2001): article B46f.
35 Two Polytope Theorem Pak, I. “Hook Length Formula and Geometric Combinatorics.” Séminaire Lotharingien de Combinatoire 46 (2001): article B46f.
36 Connection to RSK

Special Cases

Pak, I. “Hook Length Formula and Geometric Combinatorics.” Séminaire Lotharingien de Combinatoire 46 (2001): article B46f.
37 Duality

Number of Involutions in Sn

Pak, I. “Hook Length Formula and Geometric Combinatorics.” Séminaire Lotharingien de Combinatoire 46 (2001): article B46f.
38 Direct bijective Proof of the Hook Length Formula Novelli, J. C., I. Pak, and A. V. Stoyanovsky. “A direct bijective proof of the hook-length formula.” Discrete Mathematics and Theoretical Computer Science 1 (1997): 53-67.
39 Introduction to Tilings

Thurston’s Theorem

Thurston, W. P. “Conway’s tiling groups.” Amer Math Monthly 97, no. 8 (1990): 757-773.

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