Many parts of Graph Theory have witnessed a huge growth over the last years, partly because of their relation to Theoretical
Computer Science and Statistical Physics. These connections arise because graphs can be used to model many diverse structures.
The focus of this proposal is on asymptotic results, i.e. the graphs under consideration are large. This often unveils patterns and connections which remain obscure when considering only small graphs. It also allows for the use of powerful techniques such as probabilistic arguments, which have led to spectacular new developments. In particular, my aim is to make decisive progress on central problems in the following 4 areas:
Factorizations of graphs can be viewed as partitions of the edges of a graph into simple regular structures. They have a rich history and arise in many different settings, such as edge-colouring problems, decomposition problems and in information theory. They also have applications to finding good tours for the famous Travelling salesman problem.
A Hamilton cycle is a cycle which contains all the vertices of the graph. One of the most fundamental problems in Graph Theory/Theoretical Computer Science is to find conditions which guarantee the existence of a Hamilton cycle in a graph.
This is a natural (but difficult) continuation of the previous question where the aim is to embed more general structures than Hamilton cycles - there has been exciting progress here in recent years which has opened up new avenues.
In many cases, it is important to know whether a graph `strongly' possesses some property, i.e. one cannot destroy the property by changing a few edges. The systematic study of this notion is a new and rapidly growing field.