MA314-6-SP-CO:
Graph Theory

The details
2020/21
Mathematical Sciences
Colchester Campus
Spring
Undergraduate: Level 6
Current
Sunday 17 January 2021
Friday 26 March 2021
15
16 July 2020

 

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

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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 students’ 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 (Hall’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 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. Students having issues with the module are encouraged to talk to the module supervisor during office hours/open-door policy .

Bibliography*

(none)

Overall assessment

Coursework Exam
0% 100%

Reassessment

Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr David Penman, email: dbpenman@essex.ac.uk.
Dr David Penman & Dr Jesus Martinez-Garcia
Dr David Penman (dbpenman@essex.ac.uk), Dr Jesus Martinez-Garcia (jesis.martinez-garcia@essex.ac.uk)

 

Availability
Yes
Yes
No

External examiner

Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
Resources
Available via Moodle
Of 35 hours, 33 (94.3%) hours available to students:
2 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).

 

Further information
Mathematical Sciences

* Please note: due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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