# Combinatorics Available Projects

Title | Project Discription | Level | Supervisor |
---|---|---|---|

Game theory and algebraic geometry |
Nash's equilibrium is a celebrated result in mathematics and economics. It is one of the foundational results in game theory. The aim of this project is to explore the relationship between game theory and algebraic geometry. |
Honours Project | Dr Masoud Kamgarpour |

Root systems and their applications in representation theory |
Root systems are one of the most remarkable structures elucidated in 20th century mathematics. They have a simple definition in terms of linear algebra and combinatorics, but have... |
Honours Project | Dr Masoud Kamgarpour |

Projects in combinatorial geometry |
There are several research projects available in combinatorial geometry, covering areas such as polytope theory, hyperplane arrangements and triangulations, and with applications in knot theory, low-dimensional topology and operations research. Interested students are welcome to arrange a... |
PhD Project Honours Project |
Professor Benjamin Burton |

Balanced block designs |
In the latter part of the last century, balanced block designs with repetition of elements (or varieties) in the blocks started to be investigated. There remain many open problems regarding the existence and properties of such designs and their variants. |
PhD Project Masters Project |
Associate Professor Elizabeth Billington |

Graph decompositions: paths, cycles, trades |
Decompositions of complete graphs and complete multipartite graphs into cycles, paths and trails have received a fair amount of attention recently. Existence and properties of such graph decompositions, especially in the case of multipartite graphs, still need much further investigation.... |
PhD Project Masters Project |
Associate Professor Elizabeth Billington |

The biembedding of latin squares |
In broad terms the research project is based on the study of latin squares and quasigroups. More specifically latin squares may be represented as complete tripartite graphs a presentation which leads to the study of triangular embedding in the plane. To date very little is known about these... |
PhD Project | Associate Professor Diane Donovan |

Hamilton cycle decompositions of Cayley graphs and related topics |
There is an unsolved conjecture that every connected 2k-regular Cayley graph on a finite abelian group has a decomposition into k Hamilton cycles. Cayley graphs are graphs based on groups and students who like group theory or graph theory will enjoy working on this and related problems. |
PhD Project Masters Project |
Professor Darryn Bryant |

The 2-factorisation problem for complete graphs |
This project examines the existence of 2-factorisations of complete graphs in which the 2-factors are isomorphic to given 2-regular graphs. Using computers the problem has been completely solved for complete graphs of order less than 20 and several infinite families of results are known. However... |
PhD Project Masters Project |
Professor Darryn Bryant |

Perfect one-factorisations |
The perfect one-factorisation conjecture (attributed to Kotzig 1964) is that there exists a perfect one-factorisation of every complete graph of even order. To date this has only been shown to be true for two families of complete graphs: those with order one more than a prime or twice a prime. A... |
PhD Project | Dr Barbara Maenhaut |