On offset Hamilton cycles in random 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.

References in corpus (2)

• The 1-2-3 Conjecture and related problems: a survey
• On Weak Hamiltonicity of a Random Hypergraph