A Coprime Buratti-Horak-Rosa Conjecture and Grid-Based Linear Realizations

We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If $L$ is a multiset of size $v-1$ with support contained in $\{1, 2, \ldots, \lfloor v/2 \rfloor\}$ such that $\gcd(v,x) = 1$ for all $x \in L$, then $L$ is realizable. This is a specialization of the well-known BHR Conjecture and it includes Buratti's original conjecture. We argue that the most effective route to a resolution of the conjecture when the support has size 3 is to focus on $L = \{1^a, x^b, y^c\}$, where $1 (2x^2 + 2x + 1)/(x-2)$ then the Coprime BHR Conjecture holds for $\{1^a,x^b,y^c\}$ for infinitely many values of $v$, and that there are at most 3 values of $v$ for which it does not hold when $(x,y) = (6,18)$.