Tight Hamilton cycles in cherry quasirandom $3$-uniform hypergraphs

TL;DR

This paper proves the existence of tight Hamilton cycles in large cherry-quasirandom 3-uniform hypergraphs under certain minimum degree conditions using the absorbing-path method.

Contribution

It introduces new minimum degree thresholds guaranteeing tight Hamilton cycles in cherry-quasirandom 3-graphs, expanding understanding of hypergraph Hamiltonicity.

Findings

Minimum 2-degree condition ensures Hamilton cycles in large hypergraphs.

Minimum 1-degree condition with density d and alpha guarantees Hamilton cycles.

Uses absorbing-path method to establish these results.

Abstract

We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so called {\em two-path} or {\em cherry}-quasirandom $3$-graphs. Our first result asserts that for any fixed real $\alpha >0$, cherry-quasirandom $3$-graphs of sufficiently large order $n$ having minimum $2$-degree at least $\alpha (n-2)$ have a tight Hamilton cycle. Our second result concerns the minimum $1$-degree sufficient for such $3$-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every $d,\alpha >0$ satisfying $d + \alpha >1$, any sufficiently large $n$-vertex such $3$-graph $H$ of density $d$ and minimum $1$-degree at least $\alpha \binom{n-1}{2}$, has a tight Hamilton cycle.