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Course, academic year 2016/2017
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Discrete Mathematics - NDMI002
Czech title: Diskrétní matematika
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: prof. RNDr. Martin Loebl, CSc.
Mgr. Jan Kynčl, Ph.D.
Hans Raj Tiwary, M.Sc., Ph.D.
Class: Informatika Bc.
Classification: Informatics > Discrete Mathematics
Incompatibility : NDMA005
Annotation -
Last update: T_KAM (06.05.2001)

Introduction to combinatorics and graph theory. We lay stress on active knowledge of basic definitions and methods (relation, mapping, graph, exact formulation of mathematical theorems, problem solving and proofs of simple statements).
Literature -
Last update: doc. Mgr. Milan Hladík, Ph.D. (22.11.2012)

J. Matousek, J. Nesetril: Invitation to Discrete Mathematics, Oxford University Press, 2008, 2nd edition.

Syllabus -
Last update: doc. Mgr. Milan Hladík, Ph.D. (23.11.2012)

  • Basic notation, motivating examples, the concept of a proof, proof by induction.
  • Mappings, relations, equivalences. Permutations.
  • Basic combinatorial counting (the number of subsets, of subsets of size k, of all mappings, of all injective mappings, of permutations). The binomial theorem.
  • Inclusion-exclusion formula, applications (hatcheck lady).
  • Probability space (at most countable, all subsets are events). Independent events, conditional probability. Random variable, distribution function. Expectation, examples of calculation.
  • Basic notions of graph theory, path/circuit/walk, isomorphism, etc.
  • Characterisation of Eulerian graphs (including directed case; strong and weak connectedness).
  • Trees (various characterisations, existence of a leaf).
  • Planar graphs, Euler's formula, maximum number of edges.
  • The chromatic number of a graph, characterisation of bipartite graphs, chromatic number of a d-degenerate graph is at most d+1; 5-colorability of planar graphs (using Kempe's chains).
  • Partial orderings, chains and antichains, large implies tall or wide, Erdös-Szekeres lemma on monotone subsequences.

Additional topics: the HEX game, score theorem.

 
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