# MATHS 326 : Combinatorics

## Science

### Course Prescription

Combinatorics is a branch of mathematics that studies collections of objects that satisfy specified criteria. An important part of combinatorics is graph theory, which is now connected to other disciplines including bioinformatics, electrical engineering, molecular chemistry and social science. The use of combinatorics in solving counting and construction problems is covered using topics that include algorithmic graph theory, codes and incidence structures, and combinatorial complexity.

### Course Overview

This course is intended for students who have enjoyed basic combinatorics and discrete mathematics presented in MATHS 254/255 and COMPSCI 225. It expands on the concepts seen in those classes: students are introduced to graph colourings, planarity, etc. Several major theorems in the area are covered: the Five-Colour Theorem, Euler's Formula, Hall's Marriage Theorem. The syllabus also includes incidence structures such as designs, projective planes. To show applicability of combinatorics to real-life problems we consider prefix codes and universal compression codes with applications to computer science, and simple games which have applications to economics and political science. After successfully completing this course on combinatorics, students will have a solid foundation in this field of growing importance, will 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.

### Course Requirements

Prerequisite: MATHS 254 or 255, or MATHS 250 and a B+ or higher in COMPSCI 225

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Display a high level of knowledge about combinatorial structures and their properties. (Capability 1 and 2)
2. Demonstrate an understanding of the applications and connections to other disciplines. (Capability 1 and 3)
3. 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 2, 4 and 5)
4. Display ability to model and solve problems. (Capability 2 and 3)

### Assessments

Assessment Type Percentage Classification
Final Exam 50% Individual Examination
Test 20% Individual Test
Coursework 30% Group & Individual Coursework
1 2 3 4
Final Exam
Test
Coursework

### Key Topics

•  Graph colourings
• Graph algorithms
• Incidence structures
• Compression codes
• Simple games

### Special Requirements

N/A

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 or Online

Lectures will be available as recordings. Other learning activities will not be available as recordings.

The activities for the course are scheduled as a standard weekly timetable.

### 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.

There is no text book, but a set of notes will be made available on Canvas. There is also 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.

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 against online source material using computerised detection mechanisms.

### Class Representatives

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

### 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 .

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 .

### 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.

Published on 17/02/2022 09:01 a.m.