Periodic Reporting for period 1 - MIDEHA (Minimum degree conditions for tight Hamilton cycles and spanning spheres)
Reporting period: 2022-09-01 to 2024-08-31
The search for Hamilton cycles in graphs and hypergraphs has received much attention in combinatorial research over the past decades. Hamilton cycles in graphs play a fundamental role in many real world problems such as meshes in computer graphics, database theory, circuit design constructions and the Traveling Sales Man problem. Since the problem of finding a Hamilton cycle is computational intractable, the `extremal’ approach has been to identify optimal minimum degree conditions. A classic example for such a result in the graph setting is Dirac’s theorem.
For hypergraphs, the concept of cycles has been generalised in several ways. A tight cycle in a k-uniform hypergraph consists of rigidly interlocked edges following a cyclic ordering: every k consecutive vertices form an edge. Tight Hamilton cycles have been studied extensively in the last twenty years and many of the developed techniques have found application in seemingly more complex problems.
A second generalization of cycles is topological and emphasises the ‘higher dimensional’ nature of hypergraphs. Consider a cycle in a graph and observe that the simplicial complex induced by the cycle’s edges is homeomorphic to the 1-dimensional sphere. By analogy, we define a sphere in a k-graph as a set of edges whose induced simplicial complex is homeomorphic to a (k-1)-dimensional sphere and it is spanning if it contains all vertices. Spheres in hypergraphs were already considered by Brown, Erdős and Sós but the systematic investigation of these structures, and more general the study of ‘higher dimensional’ combinatorics, has only taken off quite recently.
The goal of this research project is to investigate the existence of tight Hamilton cycles and spanning spheres in hypergraphs under minimum degree conditions. To this end, a new framework for embedding large structures into hypergraphs shall be developed and tested on a series of open problems in the area.
For hypergraphs, the concept of cycles has been generalised in several ways. A tight cycle in a k-uniform hypergraph consists of rigidly interlocked edges following a cyclic ordering: every k consecutive vertices form an edge. Tight Hamilton cycles have been studied extensively in the last twenty years and many of the developed techniques have found application in seemingly more complex problems.
A second generalization of cycles is topological and emphasises the ‘higher dimensional’ nature of hypergraphs. Consider a cycle in a graph and observe that the simplicial complex induced by the cycle’s edges is homeomorphic to the 1-dimensional sphere. By analogy, we define a sphere in a k-graph as a set of edges whose induced simplicial complex is homeomorphic to a (k-1)-dimensional sphere and it is spanning if it contains all vertices. Spheres in hypergraphs were already considered by Brown, Erdős and Sós but the systematic investigation of these structures, and more general the study of ‘higher dimensional’ combinatorics, has only taken off quite recently.
The goal of this research project is to investigate the existence of tight Hamilton cycles and spanning spheres in hypergraphs under minimum degree conditions. To this end, a new framework for embedding large structures into hypergraphs shall be developed and tested on a series of open problems in the area.
The project has led to 6 research papers (with collaborators) related to the topics of the proposal, covering topics such as Hamilton cycles, perfect matchings, perfect tilings and spanning spheres in deterministic and random hypergraphs. Among these contributions is a new framework for embedding large structures into hypergraphs, which the Researcher developed in the article “Tiling dense hypergraphs”. Together with Mathias Schacht and Jan Volec, an important step towards an extension of Dirac’s theorem for hypergaphs has been taken in the article “Tight Hamiltonicity from dense links of triples”. Finally, a new framework for embedding spanning spheres in hypergraphs has been developed in the article “Spanning spheres in Dirac hypergraphs” together with Freddie
Illingworth, Alp Müyesser, Oaf Parczyk and Amedeu Sgueglia.
The outcomes of the project have been presented by the Researcher in 10 conference talks (3 invited) and 5 seminar talks. Moreover, a mini-symposium on Extremal and Probablistic Combinatorics was organised together with Prof. Nina Kamčev at the 9th European Congress of Mathematics in Seville, Spain. The project also included 6 research visits in Chile, Spain and the United Kingdom, which lead to new collaborations and progress on the scientific objectives.
Beyond this, the Researcher helped to carry out a workshop, where concepts such as cycles and trees in graphs were introduced to young learners. Moreover, the Researcher also taught a mini-course on spanning structures in hypergraphs on the 1st Brazilian Summer School of Combinatorics with lecture notes available on the project website.a
Illingworth, Alp Müyesser, Oaf Parczyk and Amedeu Sgueglia.
The outcomes of the project have been presented by the Researcher in 10 conference talks (3 invited) and 5 seminar talks. Moreover, a mini-symposium on Extremal and Probablistic Combinatorics was organised together with Prof. Nina Kamčev at the 9th European Congress of Mathematics in Seville, Spain. The project also included 6 research visits in Chile, Spain and the United Kingdom, which lead to new collaborations and progress on the scientific objectives.
Beyond this, the Researcher helped to carry out a workshop, where concepts such as cycles and trees in graphs were introduced to young learners. Moreover, the Researcher also taught a mini-course on spanning structures in hypergraphs on the 1st Brazilian Summer School of Combinatorics with lecture notes available on the project website.a
The project has led to progress on several open problems in extremal combinatorics. One of the main outcomes is a specific Dirac-type theorem for hypergraphs, which has been obtained by linking the problem to extremal set theory. This constitutes an important step towards the general solution of the problem as one can now use results and insights from this area.
Another important outcome is the development of a new framework for embedding large structures into hypergraphs. Over the course of several research papers, a set of new techniques was developed that allow to bypass the limitations of known methods. The use of these new techniques is not limited to the topics of the project, and therefore they are likely to find applications beyond the conducted research.
Another important outcome is the development of a new framework for embedding large structures into hypergraphs. Over the course of several research papers, a set of new techniques was developed that allow to bypass the limitations of known methods. The use of these new techniques is not limited to the topics of the project, and therefore they are likely to find applications beyond the conducted research.