Squares of Hamiltonian cycles in 3-uniform hypergraphs

TL;DR

This paper proves that 3-uniform hypergraphs with high minimum pair degree contain squared Hamiltonian cycles, advancing the understanding of hypergraph Hamiltonicity and contributing to the hypergraph Pósa-Seymour conjecture.

Contribution

It establishes a minimum degree condition ensuring the existence of squared Hamiltonian cycles in 3-uniform hypergraphs, a significant step in hypergraph Hamiltonian cycle theory.

Findings

Every 3-uniform hypergraph with minimum pair degree at least (4/5+o(1))n contains a squared Hamiltonian cycle.

Provides a new threshold for Hamiltonicity in hypergraphs related to the Pósa-Seymour conjecture.

Advances the understanding of cycle structures in hypergraphs.

Abstract

We show that every $3$-uniform hypergraph $H=(V,E)$ with $|V(H)|=n$ and minimum pair degree at least $(4/5+o(1))n$ contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the P\'osa-Seymour conjecture.