Transference for loose Hamilton cycles in random $3$-uniform hypergraphs

TL;DR

This paper extends minimum degree conditions for loose Hamilton cycles from dense to sparse random 3-uniform hypergraphs using a transference principle and the absorbing method, achieving asymptotic optimality for certain cases.

Contribution

It introduces a transference principle for loose Hamilton cycles in sparse hypergraphs and employs a novel contraction approach for absorber construction.

Findings

Results are asymptotically optimal for d=2.

Transference principle applies to sparse hypergraphs.

Uses a novel contraction method for absorbers.

Abstract

A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum $d$-degree condition guarantees the existence of a loose Hamilton cycle in a $k$-uniform hypergraph. For $k=3$ and each $d \in \{1,2\}$, the necessary and sufficient such condition is known precisely. We show that these results adhere to a `transference principle' to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of $d = 2$, our findings are asymptotically optimal.