Horak–Rosa Hamiltonian path conjecture

Let $K_v$ denote the complete graph on $\{0,1,\dots,v-1\}$, and let $\ell(H)$ be the multiset of edge-lengths of a graph $H$, with edge lengths in $K_v$ defined by $\ell(x,y)=\min(|x-y|,v-|x-y|)$. Horak–Rosa conjecture. Let $L$ be a list of $v-1$ positive integers not exceeding $\lfloor v/2\rfloor$. Then there exists a Hamiltonian path $H$ of $K_v$ such that $\ell(H)=L$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ does not exceed $v-d$. The conjecture generalizes Buratti's problem to arbitrary orders and is stated in the paper as still widely open, with partial results known.