On extremal hypergraphs for hamiltonian cycles

TL;DR

This paper investigates conditions that guarantee Hamiltonian cycles in hypergraphs, providing exact thresholds and degree-based criteria, advancing understanding of hypergraph Hamiltonicity.

Contribution

It introduces both Turán- and Dirac-type conditions for Hamiltonian cycles in hypergraphs, with new thresholds and degree-based criteria.

Findings

Exact Turán-type threshold for Hamiltonian cycles.

Sufficient Dirac-type degree condition.

Unified framework for hypergraph Hamiltonicity.

Abstract

We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Tur\'an- and Dirac-type results. While the Tur\'an-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.