# On offset Hamilton cycles in random hypergraphs

## TL;DR

This paper determines the precise probability threshold for the existence of $ ext{offset Hamilton cycles}$ in random hypergraphs, revealing new structural insights and connecting to the 1-2-3 Conjecture.

## Contribution

It establishes the sharp threshold for $ ext{offset Hamilton cycles}$ in random hypergraphs and explores their relation to the 1-2-3 Conjecture.

## Key findings

- Sharp threshold formula for $ ext{offset Hamilton cycles}$ in $H_{n,p}^{(k)}$
- Connection between offset Hamilton cycles and the 1-2-3 Conjecture
- Structural characterization of cycle existence in hypergraphs

## Abstract

An {\em $\ell$-offset Hamilton cycle} $C$ in a $k$-uniform hypergraph $H$ on~$n$ vertices is a collection of edges of $H$ such that for some cyclic order of $[n]$ every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) satisfies $|E_{i-1}\cap E_i|=\ell$ and every pair of consecutive edges $E_{i},E_{i+1}$ in $C$ satisfies $|E_{i}\cap E_{i+1}|=k-\ell$. We show that in general $\sqrt{e^{k}\ell!(k-\ell)!/n^k}$ is the sharp threshold for the existence of the $\ell$-offset Hamilton cycle in the random $k$-uniform hypergraph $H_{n,p}^{(k)}$. We also examine this structure's natural connection to the 1-2-3 Conjecture.