MATH 441, Introduction to Combinatorics I

Fall 2004

Course Information

Combinatorics is the study of discrete structures that arise in a variety of areas, in particular in other areas of mathematics, computer science, and many other areas of application. Central concerns are often to count objects having a particular property (for example, trees) or to prove that certain structures exist (for example, matchings of all vertices in a graph).

The textbook we will be using is A Course In Combinatorics (2nd Edition), by J.H. van Lint and R.M. Wilson.

Topics we will discuss in Math 441 include:

• Elementary graph theory and tree enumeration. (Chapters 1-2)
• Extremal combinatorics: graph coloring, Ramsey's Theorem, Erdös-Szekeres Theorem, Turan's Theorem, and Dirac's Theorem on Hamiltonian circuits. (Chapters 3-4)
• Transveral Theory, the König-Hall Theorem, posets, Sperner's Theorem, and network flows. (Chapters 5-7)
• The pebble game for deciding generic rigidity of bar frameworks in the plane
• Euler tours and De Bruin sequences. (Chapter 8)
• Inclusion-exclusion theory, Möbius inversion, Burnside's Lemma, generatingn functions, integer compositions. (Chapters 10, 13, 14)

There will be regular problem sets due the Friday of each week (starting September 3rd, 2004). The problems will come mostly from the book, but there will be additional problems given. The assignments will be available on-line at this web page.

There will be two prelims and a final examination. Your grade in the course will depend on these and the homework.

The homework will NOT be accepted after the end of class on the due date.