MA010 Graph Theory

The lectures will take place at 4pm every Thursday (starting on October 8) using the zoom platform; the link will be provided by e-mail. The lectures will be recorded and made available in the information system. The participants are welcome to switch off video and discuss possible privacy concerns with the lecturer.

The tutorials will run biweekly from the week starting on October 12, i.e., there are no tutorials in the first week of the term. Initially, all tutorials will be organized on-line via zoom only; it is almost certain that the tutorials will be held on-line during the whole term. There will be no recording of tutorials to encourage active participation.

More detailed information on the course can be found at the course webpage in the university information system..

Materials for the course can also be found in the university information system. (authentification required).

Exam

The format of the exam will be either a written in person exam or a remote on-line exam. The format will be determined based on the COVID situation in January and February 2021. The exam will a remote on-line exam lasting approximately 15-20 minutes. To register for an exam, it is necessary to satisfy the requirement for the course credit. The days when it is possible to take an exam are all Tuesdays and Thursdays during the exam period with the exception of February 2 and 4, which amount to a total of 10 exam slots. It is possible to register and unregister until 33 hours (midnight two days earlier) before each of the exam slots; additional technical instructions to those registered how to take the exam will be sent to those who have registered when the registration for the particular exam slot has closed.

Credit requirement

The credit (zápočet) requirement is the same as the condition for registering for an exam. To register for an exam, it is necessary to obtain 16 points from homework assignments. There will be four problem sheets, each containing 3 problems, each problem worth 3 points, i.e., 9 points per sheet. The deadlines for turning in solutions will be November 6 November 7, December 4, January 15 and February 12. Each of the problem sheets will be published at least about 2 weeks before the deadline. The solutions can be submitted using the university information system; resubmissions of solutions that have been graded are not allowed. The problems in the last problem sheet will be harder than those in the first three and those submitted by January 29 in the "early deadline" submission vault in the information system will be marked in a way that it is possible to register for any exam slot taking place in February.

Homework assignments

Lecture content

A brief summary of the topics covered during the lectures will be posted here. The recording of the lectures is available in the university information system..

Date Lecture content
8/10/2020 basic definitions: graph, specific graphs (complete graphs, cycles, paths), subgraph, graph minor, graph isomorphism, vertex degree, connectivity, connectivity components, path, walk, tree, forest
15/10/2020 Eulerian tour and Euler's Theorem on its existence, directed graphs, strong connectivity, planar vs. plane graph, face of a plane graph, boundary/facial walk, Kuratowski's Theorem (without proof)
22/10/2020 number of edges of a tree, Euler's formula, plane triangulations, number of edges of a planar graph, non-planarity of graphs with K5 minor, planar drawings with a chosen outerface
29/10/2020 vertex and edge colorings, Five Color Theorem, dual of a plane graph, equivalence of the Four Color Theorem and 3-edge-colorability of planar cubic bridgeless multigraphs
5/11/2020 Brooks' Theorem and its proof, Vizing's Theorem (without proof), list colorings, an example of a bipartite non-2-list-colorable graph and an example of a planar non-4-list-colorable graph
12/11/2020 Thomassen's Theorem on 5-list-colorability of planar graphs, equivalent definitions of a tree, a reminder of basic properties of a tree, existence of a spanning tree
19/11/2020 Minimum Spanning Tree (MST) problem - greedy algorithm and its implementation, edge-connectivity, 2-edge-connected graphs, Nash-Williams Theorem (without proof)
26/11/2020 vertex-connectivity, Max-flow Min-cut Theorem for unit edge capacity, Menger's Theorem in both vertex-connectivity and edge-connectivity versions
3/12/2020 structure of 2-connected graphs, matching, perfect matching, vertex cover, Kőnig's Theorem, Hall's Theorem
10/12/2020 structure of 3-connected graphs, proof of Kuratowski's Theorem, flow networks and Max-flow Min-cut Theorem (sketch of proof), networks with integral capacities
17/12/2020 classes P and NP, NP-complete problems, NP-completeness of 3-CNF formula satisfiability, graph 3-colorability and vertex cover, Π2-completeness of 3-list coloring (without proof)

Exercise problems

Sample solutions can be found in the university information system. (authentification required).