Dirac-Type Questions For Hypergraphs — A Survey (Or More Problems For Endre To Solve)

Abstract

In 1952 Dirac [8] proved a celebrated theorem stating that if the minimum degree 8(G) in an n-vertex graph G is at least n/2 then G contains a Hamiltonian cycle. In 1999, Katona and Kierstead initiated a new stream of research devoted to studying similar questions for hypergraphs, and subsequently, for perfect matchings. A pivotal role in achieving some of the most important results in both these areas was played by Endre Szemeredi. In this survey we present the current state-of-art and pose some open problems .

Tight Cycles in Hypergraphs

“…Regarding k-partite, k-uniform hypergraphs (which have k vertex classes and in which each edge contains exactly one vertex of each class), Rödl and Ruciński [9] proved an asymptotically approximate analogue of the result of Moon and Moser: For every δ > 0, every sufficiently large k-partite, k-uniform hypergraph with each class of size n, and such that each (k − 1)-set of vertices containing at most one vertex from each partition class is contained in at least ( 1 2 + δ)n edges, contains a tight Hamilton cycle. We extend this result to an asymptotically best possible minimum codegree bound guaranteeing the existence of tight cycles of various lengths.…”

Packing k-partite k-uniform hypergraphs

“…Other conditions, such as different notions of degree, which guarantee a perfect H-packing in a large k-graph G have also been considered; see the survey by Rödl and Ruciński [30] for a full account of these. In particular, in recent years there has been much study of the case of a perfect matching, see e.g.…”

Tight Co‐Degree Condition for Packing of Loose Cycles in 3‐Graphs

“…In this article, we will solve a problem of Rödl and Ruciński from (Problem 3.15) and extend Theorem to tilings with for . Theorem For every even integer , there is n 0 such that every 3‐graph H on , vertices with , can be tiled with .…”