Discrete Mathematics (MD Matemática Discreta) Grado en Ingeniería Informática, Grupo asignatura en inglés. Curso 2 General Information
Emmanuel Briand
1. General information
• • •
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Instructor: Emmanuel Briand (o ce B1.44, email:
[email protected]) O ce hours: see the webpage
http://emmanuel.jean.briand.free.fr.
Place and hours of the lessons:
Monday 12:40-2:30, room B2-30 (computer room). Thursday 10:40-12:30 am, room I0.10.
Material will be available on WebCT (diary, exercises). 2. Grades
There are two ways to pass the course.
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Continuous evaluation. Exams (examenes de convocatoria o cial). First session: june 22. Second session: September 10.
2.1.
Continuous assessment. • • • •
Continuous assessment consists in:
First exam (20% of the nal mark). Second exam (20 % of the nal mark). Third exam (25 % of the nal mark). Other activities of continuous assessment (for a total of 35 % of the nal mark): short tests in classroom and homework and projects to hand in.
Some more ambitious projects will be
proposed during the course. To pass the course, a nal mark of 5 over 10 is required. 2.2.
Exams.
The exam (convocatoria o cial) consists in a 3 hours theoretical exam, consisting in
problems and questions on the content of the course. 3. Calendar See table 1.
Monday
Thursday
Feb.9
Feb.12.
Feb.16
Feb.19
Feb.23
Feb.26
Mar.2
Mar.5
Mar.9
Mar.12
Mar.16
Mar.19
Mar.23
Mar.26 Holy week
Apr.6
Apr.9
Apr.13
Apr.16
Apr.20
No lesson on Apr. 23:Feria
Apr.27
Apr.31
No lesson on may 4 (counts as a mon-
May 7.
day) May 11
May 14
May 18
May 21
May 25
May 28
June 1st
No lesson on june 4 (Corpus Christi)
Table 1. Calendar.
1
4. Course content We will closely follow chapters 8 (graphs), 9 (trees) and 10 (digraphs) of Koshy's book Discrete Mathematics with applications (see the bibliography). We will however skip some sections and provide additional material on: connectivity in graphs, graph coloring and ows in networks. (1)
Introduction to Graphs.
Basic de nitions. Graph isomorphism. Basic algorithms: DFS and
BFS. (2)
Connectivity.
Connected components. Connectivity in digraphs: weak and strong connectivity.
Tarjan's algorithm for the strongly connected components. Biconnectivity and
k connectivity.
Cut vertices, cut sets, blocks.Edge connectivity, bridges. Menger's Theorem and Whitney Theorem. (3)
Trees.
Basic facts. Rooted trees, decision trees. Shortest path in weighted graphs: Dijkstra's
algorithm. Minimal spanning tree in weighted graphs: Prim and Kruskal's algorithms.
(4) (5)
Planarity. Planar graphs. Dual graph. Euler's formula. Eulerian and Hamiltonian walks. Eulerian circuits
Kuratowski Theorem. and trails.
Hamiltonian circuits and
trails. Dirac's Theorem (su cient condition for being hamiltonian). (6)
Colouring.
Vertex colouring: chromatic number, Greedy algorithm. Edge colouring: chromatic
index and Vizing Theorem. Bipartite graphs: Brooks Theorem, characterization. Matchings. (7)
Flows in networks.
Flows and cuts. Max ow/min cut Theorem. Max ow/min cut Algorithm. 5. Commented bibliography
Books:
•
Our main reference for this course is: N. Biggs,
Discrete Mathematics. Chapters 8 (Graphs), 9
(Trees), 10 (Bipartite graphs and matchings) and 11 (Digraphs, networks and ows). There are
http://encore.fama.us.es/iii/encore/record/ C__Rb1570946?lang=spi and http://encore.fama.us.es/iii/encore/record/C__Rb1041406? lang=spi.
several copies in the library. FAMA records:
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Another good reference is:
plied introduction.
R. Grimaldi,
Discrete and combinatorial mathematics : an ap-
Chapters 11, 12 and 13.
iii/encore/record/C__Rb2605752?lang=spi record/C__Rb1008819?lang=spi.
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T. Koshy,
http://encore.fama.us.es/ http://encore.fama.us.es/iii/encore/
FAMA records: and
Discrete Mathematics with applications (electronic book available for the students
of the university).
Chapters 8 (graphs), 9 (trees) and 10 (digraphs).
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An additional, but more complicated, reference is Diestel's classical book
edition).
There are several copies in the libraries of the University.
an e-book to all students
action?docID=10002196
access:
Graph Theory (2nd
It is also available as
http://0-site.ebrary.com.fama.us.es/lib/unisev/docDetail.
There you can print some pages or chapters from the menu
See also the book's website: fourth edition (see
http: http://
FAMA record:
//encore.fama.us.es/iii/encore/record/C__Rb1950242?lang=spi. Direct 0-www.sciencedirect.com.fama.us.es/science/book/9780124211803.
http://diestel-graph-theory.com/
InfoTools.
for a free preview of the
Electronic editions/Free Preview there).
Software:
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Graph Theory in SAGE
http://www.sagemath.org, www.sagemath.org/doc/reference/graphs.
Programming language C: we will try to implement some of our graph algorithms in C. For programming issues, you can refer to the notes of your 1rst year course "Fundamentos de pro-
Software de Introducción a la programación, Miguel Toro, (ch. 8 and 9) https:// sites.google.com/site/lsintroprogramacion/home and course webcast in Youtube https: //www.youtube.com/channel/UC1RbedII4QdgTyGilbfaWGQ • The ALGRAF Software, http://www.dma.fi.upm.es/gregorio/grafos/ALGRAF.html. gramación:
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