Graph Theory

The details
Mathematical Sciences
Colchester Campus
Postgraduate: Level 7
Sunday 17 January 2021
Friday 26 March 2021
16 July 2020


Requisites for this module



Key module for


Module description

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.

Module aims

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.

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 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.

Module information

The PGT syllabus, below, will be distinguished from the UG one by greater emphasis on proofs.

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.

Definition. Hall's theorem on matching in bipartite graphs: results equivalent to Hall's theorem. Brief (non-examinable) discussion of Tutte's theorem.

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.


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.

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. 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.



Overall assessment

Coursework Exam
0% 100%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr David Penman, email:
Dr David Penman & Dr Jesus Martinez-Garcia
Dr David Penman (, Dr Jesus Martinez-Garcia (



External examiner

Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
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

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