Horak–Rosa conjecture on edge-length lists of Hamiltonian paths

Let $K_v$ be the complete graph on \\{0,1,\ldots,v-1\\\}. For an edge $[x,y]$ of $K_v$, define its length by

$$\ell(x,y)=\min(|x-y|,v-|x-y|),$$

and let $\ell(H)$ be the multiset of edge-lengths of a Hamiltonian path $H$. Let $L$ be a list of $v-1$ positive integers not exceeding $\lfloor v/2\rfloor$. Horak and Rosa's conjecture. There exists a Hamiltonian path $H$ of $K_v$ with $\ell(H)=L$ if and only if, for every sublist $J$ of $L$ satisfying $J\cap(L\setminus J)=\emptyset$,

$$|J|\geq \gcd\\{v,\ell\mid \ell\in L\setminus J\\}-1.$$

This generalizes Buratti's conjecture from prime order to arbitrary $v$. The paper also gives an equivalent divisor formulation in the abstract and proves the conjecture when all elements of $L$ belong to $\\{1,2,3,5\\}$; the full assertion remains open in the source.