Graph Theory

Course guarantor:

Marie Demlová

Lecturer:

Marie Demlová

Tutor:

Marie Demlová

Supervisor:

Department of Mathematics

Synopsis:

Basic course in graph theory. Trees, their characterization, minimal spanning tree. Strongly connected components, rooted trees. Shortest paths, Floyds algorithm. Euler graphs and their applications, Hamiltonian graphs and their applications. Chvatal's theorem. Flow in networsk, admissible flows and admissible circulations. Matchings in general graphs and in bipartite graphs. Vertex cover and independent sets. Cliques. Colorings. Plannar graphs. Graphs and vector spaces. The content of the course is modified according to the needs of students.

Requirements:

No.

Syllabus of lectures:

1. Basic notions and properties concerning undirected graphs.

2. Vertex connectivity a edge connectivity, cut vertices, bridges.

3. Basic notions and propertis concerning directed graphs.

4. Transitive closure and transitive reduction of difectec graphs, minimally strongly connected graphs.

5. Hamiltonian graphs (undirected, directed), Chvatal's theorem.

6. Flow in networks. Ford-Fulkerson Theorem. Bases of fast algorithms for finding maximal flow in a network.

7. Admissible flows and addmisible circulations.

8. Matching in general graphs and in bipartite graphs.

9. Independent sets, cliques, vertex covers.

10. Edge covers. Colorability with emphasis to vertex colorings.

11. Graphs and vector spaces. Circiut vector subspace and cut vector subspace.

12. Duality. Duality based on plannar graphs is the same notions as duality

arising from circuits and cuts.

Syllabus of tutorials:

Excerices have the form of homeworks with consultations.

Study Objective:

The main goal of the course is to introduce to students methods and techniques used in graph theory. Emphasis is given to self study; during semestr students get small task to be solved and the solution correctly written together with its justification.

Study materials:

1. Reinhard Diestel: Graph Theory. Springer-Verlag, New York, 1997.

2. M.N.S. Swamy, K. Thulasiraman: Graphs, Networks, and Algorithms, Part 1, Graph Theory, John Wiley & Sons, New York, 1981

Note:

Further information:

https://moodle.fel.cvut.cz/courses/XP01TGR

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Wed	roomT2:C2-83 Demlová M. 08:15–09:00 (lecture parallel1 parallel nr.101) Dejvice cvičebnaroomT2:C2-83 Demlová M. 09:15–10:45 (lecture parallel1) Dejvice cvičebna
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Doctoral studies, daily studies (compulsory elective course)
• Doctoral studies, combined studies (compulsory elective course)
• Doctoral studies, structured daily studies (compulsory elective course)
• Doctoral studies, structured combined studies (compulsory elective course)