Decompositions and designs

A graph is a mathematical object which can be visualised as a collection of dots (vertices) together with a collection of lines (edges) which join some of the pairs of vertices. This versatile notion can be used to model many real-life situations involving connections or relationships between objects. For example, one could produce a graph to model a social networking platform, where vertices represent users, and edges signify friendship between two users.
A cycle is a graph in which you can start at one vertex, travel along an edge to another vertex, and keep travelling along edges until you reach the vertex you started with (with no other vertices or edges). A Hamilton cycle of a graph is a cycle contained in that graph which uses all of the vertices.
A popular research question asks how many Hamilton cycles must be contained in a graph in which every vertex sees at least half the other vertices in an edge (a Dirac graph). Together with a small group of researchers, I sought to investigate this question in the context of hypergraphs, which are a natural generalisation of graphs. We used innovative analysis of a `random walk' to study this problem, in which we imagine ourselves starting at an arbitrary vertex of the hypergraph, and flipping a complicated coin to decide what vertex to move to next, along an edge.
These techniques allowed us to prove the new result that Dirac hypergraphs indeed contain very many Hamilton cycles (or at least, the hypergraph analogue thereof), and actually, essentially as many as possible.
Together with another small group of researchers, I am currently investigating a problem related to Andersen's Conjecture, which posits that any graph which has all possible edges present and coloured in a special but natural way, must contain a `path' which uses almost all the vertices and is rainbow (has different colours on every edge). To tackle this problem, we are developing the technique of `switchings', where one makes small local changes to one graph to turn it into a different graph, with the aim to study the probabilities of certain events concerning randomly picking these graphs. We are also using new and exciting techniques involving turning our graphs into hypergraphs in order to ensure that some process we are performing in our original graph runs `efficiently' in some sense. Together, these techniques have allowed us to prove the new result that almost all of these specially coloured graphs actually contain a rainbow path which uses all vertices of the graph.
We are also seeking to apply these ideas to a question concerning slightly different rainbow structures inside other specially coloured graphs.