Description:

The goal of the class is to introduce the most important topics in graph theory, combinatorics, combinatorial structures, discrete models and algorithms. The emphasis will be not only on undestanding the basic principles but also on applications in problem solving and algorithm design. The topics include: generating functions, selected topics from graph and hypergraph coloring, Ramsey theory, introduction to probabilistic method, properties of various special classes of graphs and combinatorial structures. The theory will be also applied in the fields of combinatorics on words, formal languages and bioinformatics.

Contents:

List of the topics
1. Generating functions
2. Graph coloring and perfect graphs
3. Introduction to Ramsey theory
4. Matching in general graphs
5. Counting spanning trees
6. Introduction to probabilistic method
7. Extremal combinatorics
8. Planar graphs and Kuratowski theorem
9. Coloring graphs on surfaces
10. List coloring and choosability
11. Edge coloring
12. Combinatorial games

Seminar contents:

1. Generating functions
2. Graph coloring and perfect graphs
3. Introduction to Ramsey theory
4. Matching in general graphs
5. Counting spanning trees
6. Introduction to probabilistic method
7. Extremal combinatorics
8. Planar graphs and Kuratowski theorem
9. Coloring graphs on surfaces
10. List coloring and choosability
11. Edge coloring
12. Combinatorial games

Recommended literature:

1. B. Bollobas : Modern Graph Theory. Springer, 1998. ISBN 0-387-98488-7.
2. Graham, R. L. - Knuth, D. - Patashnik, O. : Concrete Mathematics. Addison-Wesley, 1994. ISBN 978-0-201-55802-9.
3. Diestel, R. : Graph Theory. Springer, 2016. ISBN 978-3-662-53621-6.

Keywords:

Graph, combinatorics, generating function, graph coloring, spanning trees, Ramsey theory, extremal theory, planar graphs, perfect graphs, impartial combinatorial games, graph algorithms.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar