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.
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.
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.
Excerices have the form of homeworks with consultations.
1. Reinhard Diestel: Graph Theory. Springer-Verlag, New York, 1997.