Final Report Summary - PROEXGRA (Problems in Extremal Graph Theory)
Extremal graph theory is one of the most central branches of modern combinatorial theory. It has witnessed a huge growth over the last years, partly due to its connections to several other areas of Mathematics, but also to other sciences such as Theoretical Computer Science and Statistical Physics. Graphs allow us to model real life situations such as social networks, telecommunication networks or road networks. In order to fully understand which properties of graphs are the most relevant, one has to investigate the interplay between different parameters and properties of graphs. This is the core of extremal graph theory. The aim of the project was to make decisive progress on some important problems in this area.
Some of the most natural questions in extremal graph theory ask which conditions force the existence of certain subgraphs, i.e. which conditions on a graph G guarantee an embedding of a given graph H into G. It is often very difficult to characterise all those graphs G which contain (an embedding of) H – one indication of this is that usually the corresponding algorithmic problem is NP-complete. So a very fruitful research direction has been to search for simple sufficient conditions which guarantee an embedding of H in G.
The area of embedding problems in dense graphs has seen numerous spectacular successes in recent years, many of these rely on Szemerédi's regularity lemma. This lemma asserts that any large graph can be approximated by the union of a few quasirandom graphs. A quasirandom graph is any graph that (roughly) has the same statistics as a randomly generated graph would normally have. The regularity lemma enabled mathematicians to understand the behavior of dense graphs and has turned into a powerful tool in this area.
However, much less is known for embedding problems in sparse graphs, and the development of suitable methods to tackle such problems is certainly one of the most important tasks of extremal graph theory today.
The most important output of the project was the work on an approximate solution of the so-called Loebl-Komlós-Sós conjecture for sparse graphs. This conjecture from 1995 gives a sharp condition on the median degree of a graph which forces a copy of a given tree. The central tool in the proof of the approximate version of the Loebl-Komlós-Sós conjecture is a novel decomposition method, which applies to sparse graphs too. In the case of dense graphs this decomposition produces a Szemerédi regularity partition. This new decomposition gives great insight into the structure of sparse graphs, and we expect it to have further applications. The proof of the result comprises about 160 pages, which reflects the complexity of the project.
The most fundamental result in extremal graph theory is Turán's Theorem. A special case of Turán's theorem is Mantel's theorem, which determines the maximum density of a triangle-free graph on a given number n of vertices. This leads to the natural question about the number of triangles in n-vertex graphs G of given density (above this threshold). More precisely, one can ask for the number of vertex-disjoint triangles, the number of edge-disjoint triangles and for the total number of triangles which such a graph G must have. A celebrated result by Razborov gives an asymptotic answer for the latter question. The number of edge-disjoint triangles was studied by Györi (for a certain density range). We have been able to answer the corresponding question on the number of vertex-disjoint triangles (which was already raised by Erdős). Initial steps towards such a result were made by Moon in 1968, but no further progress had been made until our complete resolution 45 years later.
Another area of the project concerns so-called graph-packings. The study of combinatorial packing and decomposition problems has a long and rich history. To date, a major branch of Combinatorics, called design theory, is devoted to this topic. Two beautiful long-standing conjectures in the area, due to Gyárfás from 1976 and Ringel from 1963, are concerned with the case when we want to pack trees into complete graphs. There had been numerous partial results on these conjectures, but they do not come any close to obtaining the full result. We have obtained an asymptotic solution of both conjectures under the additional condition that the trees have bounded degree. The proof is based on a novel `semi-random' tree embedding process.
A central problem in extremal graph theory is the Caccetta–Häggkvist conjecture, which gives a minimum outdegree condition for an oriented graph to contain a cycle of at most a given length. Despite much work over the years by a large number of researchers, even the case of finding a triangle remains open. The question of which semi-degree forces a cycle of prescribed length is equally natural. We have given an approximate answer to this question.
Scientifically the project was highly successful. It has resulted in eight publications. Originally, the project consisted of three interconnected strands: embedding problems, packing problems and Ramsey theory. Due to its complexity, the subproject on the Loebl-Komlós-Sós conjecture for sparse graphs took more time than originally anticipated. As mentioned before, we regard this as the most important output of the project, and expect that methods developed in its proof will have further applications. However, this meant that the objectives on Ramsey theory could not be completed within the duration of the project. After terminating the Marie Curie Fellowship, the fellow has started a research position at the University of West Bohemia, as a member of the EU-funded NTIS project.
Some of the most natural questions in extremal graph theory ask which conditions force the existence of certain subgraphs, i.e. which conditions on a graph G guarantee an embedding of a given graph H into G. It is often very difficult to characterise all those graphs G which contain (an embedding of) H – one indication of this is that usually the corresponding algorithmic problem is NP-complete. So a very fruitful research direction has been to search for simple sufficient conditions which guarantee an embedding of H in G.
The area of embedding problems in dense graphs has seen numerous spectacular successes in recent years, many of these rely on Szemerédi's regularity lemma. This lemma asserts that any large graph can be approximated by the union of a few quasirandom graphs. A quasirandom graph is any graph that (roughly) has the same statistics as a randomly generated graph would normally have. The regularity lemma enabled mathematicians to understand the behavior of dense graphs and has turned into a powerful tool in this area.
However, much less is known for embedding problems in sparse graphs, and the development of suitable methods to tackle such problems is certainly one of the most important tasks of extremal graph theory today.
The most important output of the project was the work on an approximate solution of the so-called Loebl-Komlós-Sós conjecture for sparse graphs. This conjecture from 1995 gives a sharp condition on the median degree of a graph which forces a copy of a given tree. The central tool in the proof of the approximate version of the Loebl-Komlós-Sós conjecture is a novel decomposition method, which applies to sparse graphs too. In the case of dense graphs this decomposition produces a Szemerédi regularity partition. This new decomposition gives great insight into the structure of sparse graphs, and we expect it to have further applications. The proof of the result comprises about 160 pages, which reflects the complexity of the project.
The most fundamental result in extremal graph theory is Turán's Theorem. A special case of Turán's theorem is Mantel's theorem, which determines the maximum density of a triangle-free graph on a given number n of vertices. This leads to the natural question about the number of triangles in n-vertex graphs G of given density (above this threshold). More precisely, one can ask for the number of vertex-disjoint triangles, the number of edge-disjoint triangles and for the total number of triangles which such a graph G must have. A celebrated result by Razborov gives an asymptotic answer for the latter question. The number of edge-disjoint triangles was studied by Györi (for a certain density range). We have been able to answer the corresponding question on the number of vertex-disjoint triangles (which was already raised by Erdős). Initial steps towards such a result were made by Moon in 1968, but no further progress had been made until our complete resolution 45 years later.
Another area of the project concerns so-called graph-packings. The study of combinatorial packing and decomposition problems has a long and rich history. To date, a major branch of Combinatorics, called design theory, is devoted to this topic. Two beautiful long-standing conjectures in the area, due to Gyárfás from 1976 and Ringel from 1963, are concerned with the case when we want to pack trees into complete graphs. There had been numerous partial results on these conjectures, but they do not come any close to obtaining the full result. We have obtained an asymptotic solution of both conjectures under the additional condition that the trees have bounded degree. The proof is based on a novel `semi-random' tree embedding process.
A central problem in extremal graph theory is the Caccetta–Häggkvist conjecture, which gives a minimum outdegree condition for an oriented graph to contain a cycle of at most a given length. Despite much work over the years by a large number of researchers, even the case of finding a triangle remains open. The question of which semi-degree forces a cycle of prescribed length is equally natural. We have given an approximate answer to this question.
Scientifically the project was highly successful. It has resulted in eight publications. Originally, the project consisted of three interconnected strands: embedding problems, packing problems and Ramsey theory. Due to its complexity, the subproject on the Loebl-Komlós-Sós conjecture for sparse graphs took more time than originally anticipated. As mentioned before, we regard this as the most important output of the project, and expect that methods developed in its proof will have further applications. However, this meant that the objectives on Ramsey theory could not be completed within the duration of the project. After terminating the Marie Curie Fellowship, the fellow has started a research position at the University of West Bohemia, as a member of the EU-funded NTIS project.