## MATH3032 Graph Theory

### 10 creditsClass Size: 200

Module manager: Dr J Britnell
Email: j.r.britnell@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisites

 MATH2210 Introduction to Discrete Mathematics

This module is approved as an Elective

### Module summary

This module provides an introduction to the basic ideas such as connectedness, trees, planar graphs, Eulerian and Hamiltonian graphs, directed graphs and the connection between graph theory and the four colour Problem. Graph theory is an important mathematical tool in such different areas as linguistics, chemistry and, especially, operational research. This module will include some abstract proofs.

### Objectives

To introduce students to some of the main concepts of graph theory.

On completion of this module, students should be able to:
(a) identify basic examples of isomorphic and non-isomorphic pairs of graphs, and make simple deductions involving vertex degrees;
(b) apply a selection of criteria related to Eulerian and Hamiltonian graphs;
(c) explain and apply the basic theories for trees, planar graphs and directed graphs;
(d) show a basic knowledge of graph colourings, and apply a range of techniques for identifying chromatic numbers for graphs and surfaces.

### Syllabus

Topics chosen from:
1. Basic definitions. Adjacency matrices, connected graphs, vertex degrees.
2. Eulerian graphs and applications.
3. Hamiltonian graphs. Dirac's theorem.
4. Trees. Cayley's Theorem.
5. Planar graphs. Euler's theorem, Kuratowski's theorem (without proof).
6. Digraphs. Robbins' Theorem, tournaments.
7. Graph colourings. The five-colour theorem for planar graphs, the four-colour theorem for planar graphs (without proof). Brook's Theorem.
8. Chromatic numbers of surfaces, Heawood's inequality.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Office Hour Discussions 20 1.00 20.00 Lecture 20 1.00 20.00 Private study hours 60.00 Total Contact hours 40.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated