MA314-7-SP-CO:
Graph Theory
2020/21
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Spring
Postgraduate: Level 7
Current
Sunday 17 January 2021
Friday 26 March 2021
15
26 February 2021
Requisites for this module
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This module introduces graph theory and some key definitions, proofs and proof techniques associated with graph theory. It is distinguished from the undergraduate version of the module by greater emphasis on the concept of proof.
1. To introduce students to basic definitions and results of graph theory.
2. To develop students’ advanced understanding of some main proofs in graph theory.
3. To develop understanding of how to apply such results in particular problems at an advanced level.
1. Give basic graph theoretic definitions.
2. Prove basic results in the theory of graphs.
3. Show critical understanding of some results about matchings (Hall’s theorem and equivalent results).
4. Have critical understanding of some basic results about Hamilton cycles.
5. Have experience of problems connected with chromatic number, and know basic theory and how to apply it.
6. Have critical understanding of extremal graph theory, Ramsey theory and the theory of random graphs.
7. Advanced understanding of proofs of key theorems in the theory of graphs.
The PGT syllabus, below, will be distinguished from the UG one by greater emphasis on proofs.
Basics
Basic definitions: degrees, walks, trails, trees: minimum spanning tree. Bipartite graphs. Cycles, Hamiltonian cycles, connectedness, connectivity. inequalities between various measures of connectivity. Minors, subdivisions. Planar graphs (not too much topological detail). Independent sets and cliques, complements.
Matchings
Definition. Hall's theorem on matching in bipartite graphs: results equivalent to Hall's theorem. Brief (non-examinable) discussion of Tutte's theorem.
Connectivity
Menger's theorem, related results.
Hamilton cycles
Dirac's theorem, Ore's theorem. Chvatal-Erdos theorem. Hamiltonicity or otherwise of simple graphs such as the Petersen graph.
Colouring
Chromatic number. Greedy colouring algorithm, statement of Brooks' theorem. Edge chromatic number: Vizing's theorem. Choosability: Galvin's theorem (without proof). Brief informal discussion of the four-colour problem (non-examinable). Proofs of some appropriate results.
Extremal Graph Theory
Turan's theorem (with proof). Erdos-Stone theorem (without proof). Examples including proofs of simple results. .
Ramsey theory
Basic definitions. Erdos-Szekeres upper bound: probabilistic lower bound on diagonal Ramsey numbers. Some calculations for small parameters and proofs of basic results.
Linear Algebra Methods
Adjacency matrix and spectrum. Some links with graph structure. Strongly regular graphs and their eigenvalues. Multiplicities of their eigenvalues and examples of using this.
Random graphs
G(n,p). Thresholds. First and second moment methods. Simple examples.
Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.
Specific to the PGT students there will be additional lectures emphasising proof.
Students having issues with the module are encouraged to talk to the module supervisor during office hours/open-door policy.
This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Exam |
Main exam: 180 minutes during Summer (Main Period)
|
Additional coursework information
Formative problem sheets will be provided, and discussed in some of the lectures.
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Jesus Martinez-Garcia, email: jesus.martinez-garcia@essex.ac.uk.
Dr Jesus Martinez-Garcia & Dr Dan Brawn
Dr Jesus Martinez-Garcia (jesus.martinez-garcia@essex.ac.uk), Dr Dan Brawn (dbrawn@essex.ac.uk)
No
No
No
Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 2206 hours, 0 (0%) hours available to students:
2206 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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