## MA314-6-SP-CO:

Graph Theory

## Key module for

## Module description

## Module aims

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

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

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

## Bibliography

- Wilson, R.J. (2010)
*Introduction to graph theory*. 5th ed. Harlow: Pearson Education. Available at: https://ebookcentral.proquest.com/lib/universityofessex-ebooks/detail.action?docID=5136626.

The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

Further reading can be obtained from this module's reading list.

## 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.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

0% | 100% |

### Reassessment

Coursework | Exam |
---|---|

0% | 100% |

## External examiner

**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.

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