## Final Report Summary - APGRAPH (Asymptotic Graph Properties)

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 project was on asymptotic results, i.e. the graphs under consideration are large. This approach 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, project led to significant progress on central problems in the following four areas:

Designs and Factorizations of graphs: these can be viewed as partitions of the edges of a graph (or hypergraph) into simple structures. They have a rich history and arise in many different settings, such as edge-colouring problems, scheduling problems, statistical planning, decomposition problems and in information theory. Highlights obtained within the project include a new proof of the existence of designs (based on purely combinatorial techniques), a proof of the existence of F-designs for arbitrary hypergraphs F, the resolution of the Oberwolfach conjecture, a proof of the 1-factorization conjecture, an approximate version of the Erdos conjecture on locally sparse Steiner triple systems and a proof of the tree packing conjecture for bounded degree trees.

2. Hamilton cycles: 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. We made substantial advances, e.g. by settling a problem from the 1970’s on the existence of Hamilton cycles in dense regular graphs and by resolving the Hamilton decomposition conjecture.

3. Embeddings of graphs: This is a natural (but difficult) continuation of the previous question where the aim is to embed more general structures than Hamilton cycles. We developed a powerful embedding tool which finds an almost optimal packing of bounded degree graphs within a quasi-random graph. This in turn was a key tool for several results obtained in Part 1 of the project.

4. Resilience of graphs and property testing: In many cases, it is important to know whether a graph `strongly' possesses some property P, i.e. one cannot destroy the property by changing a few edges. The systematic study of this notion is a recent and rapidly growing field, to which the project also contributed (e.g. for the case where P is the property of being Hamiltonian). On the other hand, it may be helpful to find out quickly if a graph is at least `close’ to satisfying some property - a notion which is known as testability. We were able to obtain a characterization of all hypergraphs which are efficiently testable.

The focus of this project was on asymptotic results, i.e. the graphs under consideration are large. This approach 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, project led to significant progress on central problems in the following four areas:

Designs and Factorizations of graphs: these can be viewed as partitions of the edges of a graph (or hypergraph) into simple structures. They have a rich history and arise in many different settings, such as edge-colouring problems, scheduling problems, statistical planning, decomposition problems and in information theory. Highlights obtained within the project include a new proof of the existence of designs (based on purely combinatorial techniques), a proof of the existence of F-designs for arbitrary hypergraphs F, the resolution of the Oberwolfach conjecture, a proof of the 1-factorization conjecture, an approximate version of the Erdos conjecture on locally sparse Steiner triple systems and a proof of the tree packing conjecture for bounded degree trees.

2. Hamilton cycles: 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. We made substantial advances, e.g. by settling a problem from the 1970’s on the existence of Hamilton cycles in dense regular graphs and by resolving the Hamilton decomposition conjecture.

3. Embeddings of graphs: This is a natural (but difficult) continuation of the previous question where the aim is to embed more general structures than Hamilton cycles. We developed a powerful embedding tool which finds an almost optimal packing of bounded degree graphs within a quasi-random graph. This in turn was a key tool for several results obtained in Part 1 of the project.

4. Resilience of graphs and property testing: In many cases, it is important to know whether a graph `strongly' possesses some property P, i.e. one cannot destroy the property by changing a few edges. The systematic study of this notion is a recent and rapidly growing field, to which the project also contributed (e.g. for the case where P is the property of being Hamiltonian). On the other hand, it may be helpful to find out quickly if a graph is at least `close’ to satisfying some property - a notion which is known as testability. We were able to obtain a characterization of all hypergraphs which are efficiently testable.