## Final Report Summary - QRGRAPH (Quasirandomness in Graphs and Hypergraphs)

Combinatorics is a branch of mathematics focusing on the study of finite structures, such as networks, which are both fundamental mathematical objects and ubiquitous in everyday life. One of the main areas of Combinatorics is Graph Theory. Graphs consist of vertices (which can be thought of as points) some of which are joined by edges (which can be thought of as lines). Graphs are very simple mathematical structures, but give rise to incredibly complex problems, that are often beyond the capacity of current computers to solve. They can be used to model many real-life situations, e.g. social networks, biological networks and structures in computer science and communication theory. In recent years powerful new techniques have been developed, which have led to tremendous growth and a number of spectacular successes in the field.

One of these techniques is the use of quasirandom decompositions. Roughly speaking, a structure is called quasirandom if it has a number of properties that one would expect from a random structure with similar parameters. For instance, a graph is quasirandom if its edges are spread evenly over the vertices (i.e. no subgraph is significantly denser or sparser than the graph itself). This concept has been remarkably useful in many areas of Mathematics and Computer Science. For example, in Number Theory, it was first used by Szemerédi to prove the existence of long arithmetic progressions in arbitrary dense sets of the integers in 1975.

Quasirandomness is a field that is developing very rapidly, but there are many connections and properties that are still unexplored. The aim of the project was to use quasirandomness in order to tackle various important open problems Combinatorics/Graph Theory, thereby developing the method further.

One class of problems studied in the project, where quasirandomness was crucial for further progress, are matching problems. A perfect matching in a graph is a set of disjoint edges which together cover all the vertices. Many problems can be formulated as matching problems. For instance, given a set of people, construct a graph by drawing an edge if they like each other – a perfect matching splits the people into teams of two which can work together. How and when this can be achieved for teams of two is well understood. The situation changes when trying to arrange teams of three or more. This can be formulated as a hypergraph matching problem. A major open problem in the area is to determine the minimum vertex degree which guarantees a matching of given size in a uniform hypergraph. We were able to make significant progress on this problem and several related questions.

Another class of problems studied in the project are decomposition problems for graphs. An important example of such a problem asks for a Hamilton decomposition of a graph. To decide whether a graph contains a Hamilton cycle is one of the most fundamental problems in the area (this is also very difficult from an algorithmic point of view) – here a Hamilton cycle in a graph is a cycle containing all vertices of the graph. A far stronger requirement is to ask for a Hamilton decomposition of a graph G: this is a set of Hamilton cycles decomposing all the edges of G – so every edge is contained in exactly one of the Hamilton cycles. Hamilton decompositions are notoriously hard to obtain and so there are many open problems in this area – two of the most famous of these were Kelly's conjecture from 1968 on Hamilton decompositions of tournaments and the Hamilton decomposition conjecture of Nash-Williams from 1970. There was also a related conjecture (the so-called 1-factorization conjecture) on decompositions into perfect matchings. (This is related to edge-colourings of graphs.) We were able to prove all three of these conjectures. The methods we developed in these proofs have many further applications, e.g. to the well known Travelling salesman problem.

A further class of problems studied in the project are concerned with subdivisions of graphs. This is a fundamental graph theoretical concept. For example, the class of all graphs which can be drawn in the plane without crossings can be characterized by forbidden subdivisions. There has been much research in this area, but many tantalizing open problems remain. One of these was solved within the project, settling a longstanding conjecture of Mader.

One of these techniques is the use of quasirandom decompositions. Roughly speaking, a structure is called quasirandom if it has a number of properties that one would expect from a random structure with similar parameters. For instance, a graph is quasirandom if its edges are spread evenly over the vertices (i.e. no subgraph is significantly denser or sparser than the graph itself). This concept has been remarkably useful in many areas of Mathematics and Computer Science. For example, in Number Theory, it was first used by Szemerédi to prove the existence of long arithmetic progressions in arbitrary dense sets of the integers in 1975.

Quasirandomness is a field that is developing very rapidly, but there are many connections and properties that are still unexplored. The aim of the project was to use quasirandomness in order to tackle various important open problems Combinatorics/Graph Theory, thereby developing the method further.

One class of problems studied in the project, where quasirandomness was crucial for further progress, are matching problems. A perfect matching in a graph is a set of disjoint edges which together cover all the vertices. Many problems can be formulated as matching problems. For instance, given a set of people, construct a graph by drawing an edge if they like each other – a perfect matching splits the people into teams of two which can work together. How and when this can be achieved for teams of two is well understood. The situation changes when trying to arrange teams of three or more. This can be formulated as a hypergraph matching problem. A major open problem in the area is to determine the minimum vertex degree which guarantees a matching of given size in a uniform hypergraph. We were able to make significant progress on this problem and several related questions.

Another class of problems studied in the project are decomposition problems for graphs. An important example of such a problem asks for a Hamilton decomposition of a graph. To decide whether a graph contains a Hamilton cycle is one of the most fundamental problems in the area (this is also very difficult from an algorithmic point of view) – here a Hamilton cycle in a graph is a cycle containing all vertices of the graph. A far stronger requirement is to ask for a Hamilton decomposition of a graph G: this is a set of Hamilton cycles decomposing all the edges of G – so every edge is contained in exactly one of the Hamilton cycles. Hamilton decompositions are notoriously hard to obtain and so there are many open problems in this area – two of the most famous of these were Kelly's conjecture from 1968 on Hamilton decompositions of tournaments and the Hamilton decomposition conjecture of Nash-Williams from 1970. There was also a related conjecture (the so-called 1-factorization conjecture) on decompositions into perfect matchings. (This is related to edge-colourings of graphs.) We were able to prove all three of these conjectures. The methods we developed in these proofs have many further applications, e.g. to the well known Travelling salesman problem.

A further class of problems studied in the project are concerned with subdivisions of graphs. This is a fundamental graph theoretical concept. For example, the class of all graphs which can be drawn in the plane without crossings can be characterized by forbidden subdivisions. There has been much research in this area, but many tantalizing open problems remain. One of these was solved within the project, settling a longstanding conjecture of Mader.