## MA314-6-SP-CO:Graph Theory

The details
2022/23
Mathematical Sciences
Colchester Campus
Spring
Current
Sunday 15 January 2023
Friday 24 March 2023
15
29 April 2022

Requisites for this module
MA108 and MA114
(none)
(none)
(none)

(none)

(none)

## Module description

This module introduces graph theory and some key definitions, proofs and proof techniques associated with graph theory.

## Module aims

1. To introduce students to basic definitions of graph theory.
2. To develop studentsâ€™ critical understanding of some main results in graph theory.
3. To develop understanding of how to apply such results in particular problems.

## Module learning outcomes

1. Give basic graph theoretic definitions.
2. Prove basic results in the theory of graphs.
3. Show critical understanding of some results about matchings (Hallâ€™s theorem and equivalent results).
4. Have critical understanding of some basic results about Hamilton cycles.
5. Have some 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.

## Module information

Syllabus:

Basics
Basic definitions: degrees, walks, trails, trees: minmum 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. Choosabiilty: Galvin's theorem (without proof). Brief informal discussion of the four-colour problem (non-examinable).

Extremal Graph Theory

Turan's theorem (with proof). Erdos-Stone theorem (without proof). Examples.

Ramsey theory
Basic definitions. Erdos-Szekeres upper bound: probabilistic lower bound on diagonal Ramsey numbers. Some calculations for small parameters.

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 .

## Learning and teaching methods

Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography

The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

## Assessment items, weightings and deadlines

Coursework / exam Description Deadline Coursework weighting
Exam  Main exam: In-Person, Closed Book, 120 minutes during Summer (Main Period)
Exam  Reassessment Main exam: In-Person, Closed Book, 120 minutes during September (Reassessment Period)

### 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.

Coursework Exam
0% 100%

### Reassessment

Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr Alexei Vernitski, email: asvern@essex.ac.uk.
Dr Alexei Vernitski
asvern@essex.ac.uk

Availability
Yes
Yes
No

## External examiner

Prof Stephen Langdon
Brunel University London
Professor
Resources
Available via Moodle
Of 11 hours, 7 (63.6%) hours available to students:
4 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.

Further information
Mathematical Sciences

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