470-4302/03 – Graph Theory (TG)

Gurantor departmentDepartment of Applied MathematicsCredits4
Subject guarantordoc. Mgr. Petr Kovář, Ph.D.Subject version guarantordoc. Mgr. Petr Kovář, Ph.D.
Study levelundergraduate or graduate
Study languageCzech
Year of introduction2016/2017Year of cancellation
Intended for the facultiesUSPIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
KOV16 doc. Mgr. Petr Kovář, Ph.D.
KUB59 RNDr. Michael Kubesa, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2
Combined Credit and Examination 10+10

Subject aims expressed by acquired skills and competences

Each student is supposed to - analyze real life problems - express them as a graph theory problem - solve the problem using graph theory methods - give an interpretation of the theoretical results in the terms of the original problems At the same time he should decide what are the limits of an ideal theoretical solution in contrast to the real situation.

Teaching methods

Lectures
Individual consultations
Tutorials
Project work

Summary

The course covers both basic and advanced topics of Graph Theory, often overlapping with other branches of mathematics (algebra, combinatorics). A mandatory part of the course is one or sometimes two projects focused on real life problems that are solved using methods of graph theory.

Compulsory literature:

J. Matoušek, J. Nešetřil, Chapters in Discrete Mathematics, Karolinum Praha (2000).

Recommended literature:

D. B. West, Introduction to graph theory - 2nd ed., Prentice-Hall, Upper Saddle River NJ, (2001).

Way of continuous check of knowledge in the course of semester

During the semester each student prepares one or two projects. Topic of each project is chosen among topics presented in the lecture. The sumbission date and all assignments are available on the instructor webpage or in the university electronic information system. (Each student has to register for a topic.) For a pass it is necessary to have at least one project graded for at least 10 points.

E-learning

Další požadavky na studenta

There are no other requirements.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Lectures: 1) Graphs, simple graphs. Incidence matrix and adjacency matrix. Subgraphs. Degree of a vertex. 2) Paths and cycles. 3) Trees, bridges and cuts. 4) Graph isomorphisms. 5) Connectivity and blocks. Cut sets. 6) Matching and covers in general and bipartite graphs. Perfect matchings. 7) Edge colorings. Chromatic index, Vizing's Theorem. 8) Vertex colorings, Chromatic number, Brook's Theorem. 9) Planar graphs. Dual graphs, Euler's formula for connected planar graphs, Kuratowski's Theorem. Four Color Theorem. 10) Non-planar graph, measures of non-planarity. 11) Eulerian and Hamiltonian graphs. 12) Oriented graphs. Oriented paths and cycles. 13 and 14) Further topics: flows in networks, cuts. Maximal flow and minimal cut Theorem. Graph models. Discussions: 1) Graphs, simple graphs. Degree of a vertex. 2) Paths and cycles. Important graph classes. 3) Trees, bridges and cuts. 4) Graph isomorphisms. 5) Graph connectivity, blocks and articulations. 6) Matching and covers in general and bipartite graphs. Perfect matchings. 7) Edge colorings. Chromatic index. 8) Vertex colorings, Chromatic number. 9) Planar graphs. Euler's formula for general planar graphs. 10) Non-planar graphs, measures of non-planarity. 11) Eulerian and Hamiltonian graphs. 12) Oriented graphs. Oriented paths and cycles. 13 and 14) Further topics; graph models and their limits. During the semester each student prepares one or two projects. Topic of each project is chosen among topics presented in the lecture. Assignments are available on the instructor webpage or in the university electronic information system, for example: 1) planar graphs and their applications 2) graph colorings and their applications For a pass it is necessary to have at least one project graded for at least 10 points.

Conditions for subject completion

Full-time form (validity from: 2016/2017 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
        Credit Credit 20  10
        Examination Examination 80 (80) 41
                Written part Written examination 60  30
                Theory part of the exam Oral examination 20  5
Mandatory attendence parzicipation: participation at all exercises is obligatory, 2 apologies are accepted participation at all lectures is expected, 4 apologies are accepted

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Combined form (validity from: 2016/2017 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
        Credit Credit 20  10
        Examination Examination 80 (80) 41
                Written part of the exam Written examination 60  30
                Theory part of the exam Oral examination 20  10
Mandatory attendence parzicipation: participation at all exercises is obligatory, absence of 20% can be excused participation at all lectures is expected, absence of 30% can be excused

Show history

Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2018/2019 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan
2017/2018 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan
2016/2017 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner