2002/03
Undergraduate Module Catalogue
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MATH3032
Graph Theory
10 credits
Taught
Semester 1,
Year running
2002/03
Pre-requisites
MATH2210 or equivalent.
Co-requisites
None
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.
Form of teaching
Lectures: 20 hours. Other Hours: 20 office hours.
Form of assessment
One 2 hour examination at end of semester (85%). Coursework (15%).
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Errors, omissions, failed links etc should be notified to
Geoff Lidster.
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