### Discrete Mathematics 1

#### Learning Outcomes

1. To solve homogeneous and non-homogeneous linear recursion.
2. Apply the recursive way of thinking on the appropriate combinatorial problems.
3. Describe and manipulate with basic notions from graph theory.
4. Describe and solve appropriate engineering problems in terms of graph theory.
5. Solve problems algorithmically using computers.

#### Forms of Teaching

Lectures

4 hours of lectures a week. The lecturer is going to follow the lecture materials.

Independent assignments

During the semester 3 software excercises will be given to be solved by the students individually and to be shown in the framework of laboratory excercises.

#### Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Laboratory Exercises 0 % 15 % 0 % 15 %
Homeworks 0 % 5 % 0 % 5 %
Mid Term Exam: Written 0 % 40 % 0 %
Final Exam: Written 0 % 40 %
Exam: Written 0 % 80 %

#### Week by Week Schedule

1. Fibonacci's sequence; Homogeneous recursive relations.
2. Homogeneous recursive relations.
3. Non-homogeneous recursive relations.
4. Notion of a graph; Main definitions and examples; Graph isomorphism.
5. Connectivity; Paths; Eulerian and hamiltonian graphs; Optimization algorithms; Shortest path problem; Chinese postman problem; Travelling salesman problem.
6. Different characterizations of trees.
7. Labelled trees; Prüfer code; Minimum spanning tree; Algorithms.
8. Midterm exam.
9. Kuratowski's theorem; Euler's formula.
10. Dual graphs; Planarity testing algorithms.
11. Vertex colourings; Brooks' theorem; Edge colourings.
12. Face colourings of planar graphs; Four colour theorem.
13. Notion of a digraph; Connectivity and strong connectivity; Critical path problem; Algorithms; Tournaments.
14. Marriage problem; Hall's theorem; Transversals; Application to latin squares.
15. Final exam.

#### Study Programmes

##### University undergraduate
Computing (study)
(3. semester)

#### Literature

(.), D. Kovačević, D. Žubrinić, Uvod u diskretnu matematiku, Element, Zagreb, 2014.,
(.), A. Nakić, M.-O. Pavčević, Uvod u teoriju grafova, Element, Zagreb, 2014.,
(.), J.M. Aldous, R.J. Wilson, Graphs and Applications - An Introductory Approach, Springer, 2004.,
(.), G. Chartrand, L. Lesniak, Graphs and Digraphs, CRC Press, 2005.,

#### General

ID 183396
Winter semester
6 ECTS
L1 English Level
L1 e-Learning
60 Lectures
0 Exercises
6 Laboratory exercises
0 Project laboratory

85 Excellent
70 Very Good
55 Good
45 Acceptable