Discrete Mathematics

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Discrete Mathematics

Lattice models of polymer systems

Long chain polymers like DNA can be modelled by walks, polygons, trees, and various other combinatorial structures embedded in lattices. This project aims to investigate new polymer models. This can be approached using exact solution techniques or computational methods like series enumeration and random sampling.

Contact: Nick Beaton nrbeaton@unimelb.edu.au

Graph Colourings and Oriented Graphs

By defining graph colouring using graph homomorphism, we can build a definition for graph colouring for directed graphs that, in some sense, takes into account the direction of the arcs. In this analogue, our intuition for how graph colourings should be behave often is mistaken. Well understood results and bounds, like Brooks' Theorem or the Four-Colour Theorem no longer hold. In this project we will look at some subgraphs of complete graphs that, surprisingly, can only be coloured by assigning. every vertex its own colour. We examine what change is possible when we reverse the direction of a subset of arcs.

Contact: Christopher Duffy christopher.duffy@unimelb.edu.au