On offset Hamilton cycles in random hypergraphs

Let k and ℓ be integers satisfying 1≤ℓ≤k∕2. An ℓ-offset Hamilton cycleC in a k-uniform hypergraph H on n vertices is a collection of edges of H such that for some cyclic order of [n] and every even i every pair of consecutive edges Ei−1,Ei in C (in the natural ordering of the edges) satisfies |Ei−1∩Ei|=ℓ and every pair of consecutive edges Ei,Ei+1 in C satisfies |Ei∩Ei+1|=k−ℓ. We show that in general ekℓ!(k−ℓ)!∕nk is the sharp threshold for the existence of the ℓ-offset Hamilton cycle in the random k-uniform hypergraph Hn,p(k). We also examine this structure's natural connection to the 1-2-3 Conjecture.