MATHS 326 : Combinatorics
Science
2024 Semester One (1243) (15 POINTS)
Course Prescription
Course Overview
This course is intended for students who have enjoyed basic combinatorics and discrete mathematics presented in MATHS 254 and COMPSCI 225. It expands on the concepts seen in those classes: students are introduced to graph colourings, planarity and more. Several major theorems in the area are covered, for example, the Five-Colour Theorem, Euler's Formula and Hall's Marriage Theorem. The syllabus also includes incidence structures such as designs and projective planes and an introduction to discrete potential theory. To show applicability of combinatorics to real-life problems we consider prefix codes and universal compression codes with applications to computer science. After successfully completing this course on combinatorics, students will have a solid foundation in this field of growing importance and be familiar with some important applications of combinatorics. Students will also be well prepared for further graduate courses on the topic, such as MATHS 715 or MATHS 782.
Capabilities Developed in this Course
| Capability 3: | Knowledge and Practice |
| Capability 4: | Critical Thinking |
| Capability 5: | Solution Seeking |
| Capability 6: | Communication |
| Capability 7: | Collaboration |
Learning Outcomes
- Display a high level of knowledge about combinatorial structures and their properties. (Capability 3)
- Demonstrate an understanding of the applications and connections to other disciplines. (Capability 3, 4 and 5)
- Demonstrate an understanding of the nature of proof and precision in mathematics, and the ability to present arguments both formally and informally in written form. (Capability 4 and 6)
- Display their ability to model and solve problems both individually and as part of a team. (Capability 5, 6 and 7)
Assessments
| Assessment Type | Percentage | Classification |
|---|---|---|
| Final Exam | 50% | Individual Examination |
| Test | 20% | Individual Test |
| Coursework | 30% | Group & Individual Coursework |
| 3 types | 100% |
| Assessment Type | Learning Outcome Addressed | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | |||||||
| Final Exam | ||||||||||
| Test | ||||||||||
| Coursework | ||||||||||
Tuākana
Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials,
one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html
Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.
Key Topics
- Graph colourings (chromatic polynomial, 5-Colour Theorem, relations between chromatic number and other graph parameters, edge colouring)
- Graph algorithms (minimum spanning tree, shortest path, maximum matching in bipartite graphs, König's and Hall's theorems)
- Incidence structures (necessary conditions including Fisher's inequality and the Bruck-Ryser-Chowla theorem, Steiner systems, affine and projective planes, mutually orthogonal latin squares)
- Discrete potential theory (existence and uniqueness of harmonic functions with given boundary conditions, random walks and electrical networks, square tilings)
- Compression codes (Kraft's inequality, Huffman codes, Shannon information, Fitingof's code)
Workload Expectations
This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. While different students will allot their time in different ways that match their skills and preferences, a common breakdown of this time per week is as follows: 4 hours for in class activities, such as lectures and tutorials, 3 hours for working on your assessments and 3 hours for revision and studying with peers.
Delivery Mode
Campus Experience
- The activities for the course are scheduled as a standard weekly timetable.
- Attendance is required at scheduled activities including team tasks.
- Lectures will be available as recordings. Other learning activities will not be available as recordings.
Learning Resources
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
Notes will be made available on Canvas.
There is a broad collection of books in the library on these topics, as well as plenty of online resources.
Student Feedback
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
Other Information
This course uses a Team Based Learning approach. At the beginning of the course, students are assigned to teams of approximately five students in each team. After a preliminary topic devoted to revision of the basics, the course is divided into five main topics. Each topic has the same structure: preliminary reading, available in advance, followed by a readiness assurance test (RAT). This is a multi-choice quiz based on the reading, and it has individual and team components. Near the end of each topic, there is a team task, where students are set a problem to solve as a group during class.
Academic Integrity
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.
Class Representatives
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
Copyright
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
Inclusive Learning
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
Special Circumstances
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
Learning Continuity
In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Disclaimer
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.