# 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

### Hamiltonicity of vertex-transitive graphs

A question of Lovasz in 1969 asked whether every connected Cayley graph with more than two vertices have a Hamilton cycle. This question is still wide open, even for Cayley graphs on some fairly simple families of groups. More surprisingly, there are only a few examples of vertex-transitive graphs known to have no Hamilton cycle, which leads to the conjecture that there are only finitely many vertex-transitive graphs with no Hamilton cycles. The aim of this project is to investigate the Hamiltonicity for some special families of vertex-transitive graphs, especially Cayley graphs.

**Contact:** Binzhou Xia binzhoux@unimelb.edu.au