## MATH3032 Graph Theory

### 10 creditsClass Size: 200

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH2210 or equivalent.

This module is approved as an Elective

### 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

Graph theory is an important mathematical tool in such different areas as linguistics, chemistry and, especially, operational research. But its origins are in mathematical puzzles such as that of the Bridges of Kvnigsberg, and graph theory continues to have its own intellectual appeal apart from its practical applications. The module will provide 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. The module will include some abstract proofs. The homework is an essential part of the module. 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

Lectures: 20 hours. Other Hours: 20 office hours.

### Methods of assessment

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

One 2 hour examination at end of semester (85%). Coursework (15%).