# MATHS 715 : Graph Theory and Combinatorics

## Science

### Course Prescription

A study of combinatorial graphs (networks), designs and codes, illustrating their application and importance in other branches of mathematics and computer science.

### Course Overview

This course is intended for students who have enjoyed the introduction to abstract algebra and parts of discrete mathematics presented in MATHS 254, 320, 326 and 328. The main focus of the course is on combinatorial structures, especially graphs and their properties. Emphasis is placed on the contexts in which these structures occur, methods by which they can be modelled and analysed, and their diverse applications. We introduce students to a wide variety of aspects of the subject and some of its major theorems. A student completing this course will have a wide knowledge of fundamentals and approaches in this eld of growing importance.

A good background in basic linear and abstract algebra is assumed as well as some knowledge of graphs and other discrete structures (e.g. from COMPSCI 225 or MATHS 326).

### Course Requirements

Prerequisite: 15 points from MATHS 320, 326, 328 with a B or higher

### Capabilities Developed in this Course

 Capability 3: Knowledge and Practice Capability 4: Critical Thinking Capability 5: Solution Seeking Capability 6: Communication

### Learning Outcomes

By the end of this course, students will be able to:
1. Demonstrate familiarity and confidence in using the definitions and tools of graph theory and combinatorics, as well as tools from linear algebra and fundamental concepts of discrete mathematics, to answer questions and solve theoretical problems in these fields. (Capability 3, 4 and 5)
2. Derive properties of graphs and other combinatorial structures from their fundamental axioms, and draw conclusions from these properties. (Capability 3, 4 and 5)
3. Develop clear and concise proofs of important fundamental theorems in graph theory and combinatorics, and to communicate these effectively in written form. (Capability 3, 4, 5 and 6)
4. Generate and describe examples of graphs and other combinatorial structures satisfying given feasible conditions. (Capability 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Test 20% Individual Test
Final Exam 50% Individual Examination
1 2 3 4
Assignments
Test
Final Exam
The final grade for each student will be calculated using the plussage system, taking the maximum of two scores: either 30% from assignments, 20% from the test and 50% from the final exam, or 100% from the final exam.  Every student needs to score at least 35% in the final exam to pass the course.

### 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

• Basic graph theory results, including Ore's Theorem on Hamilton cycles, Cayley's Theorem on trees, and Brooks' and Vizing's Theorems on colourings
• Spectral graph theory and applications (e.g. to strongly regular graphs and Moore graphs)
• Graph automorphisms, including Cayley graphs
• Probabilistic method, including for example the union bound, Markov's inequality and some applications to extremal graph theory
• Generating functions
• Other selected topics, including extremal graph theory and abstract polytopes

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. For each week of this course, you can expect 3 hours of lectures, 3 hours of reading and thinking about the content and 4 hours of work on assignments and/or test of final exam preparation.

### Delivery Mode

#### Campus Experience

• Activities for the course are scheduled in a standard weekly timetable.
• Attendance is highly recommended at lectures, which are unlikely to be available later as recordings.
• There are no scheduled tutorials or group discussions.

### 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.

A set of course notes will be made available on Canvas.

There is also a broad collection of books in the library on the course 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.

There was no strong feedback from students in 2023 suggesting changes for 2024, but nevertheless some of the material has been updated.

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

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 31/10/2023 10:53 a.m.