Minimum degree conditions for tight Hamilton cycles and spanning spheres

Objetivo

One of the most exciting developments in the second half of the last century in combinatorial research has been the search for Hamilton cycles in graphs and hypergraphs. Since the decision problem, whether a given graph contains a Hamilton cycle, is computationally intractable, no `simple' characterization for their existence is known. The main approach to finding Hamilton cycles has thus focused on natural sufficient conditions. A classic example for this is Dirac's theorem, which provides optimal minimum degree conditions for the existence of a Hamilton cycle in graphs.

The aim of this project is to resolve several problems regarding hypergraph analogues of Dirac's theorem. The proposed research considers two natural generalization of cycles: (i) Tight cycles, which have been extensively researched in the past two decades, and (ii) Spheres, a topological generalization of cycles, which was suggested by Brown, Erdős and Sós in the Seventies and has recently resurfaced in extremal graph theory. To determine optimal minimum degree conditions for spanning tight cycles and spheres, the experienced researcher plans to develop new techniques based on hypergraph regularity and combinatorial optimization, which will likely find application beyond the proposed research.