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

Abstract

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 $1<x<y$, with $a$ large subject to $a < x+y$. We use grid-based graphs to construct linear realizations for many such multisets. A partial list of parameter sets that the constructions cover: $a = x+y-1$; $a = x+y-2$ when $x=3$ or $x$ is even; $a \geq 4x-3$ for $x$ odd, $y > 2x-2$, and $b \geq y-2x+2$; $a \geq x$ for $y=tx$, with $x$ and $t$ odd, and $b \geq tx+2t-3$; $a \geq 7$ for $x=3$ and $b \geq y-4$. As well as these (and further) immediate results, the techniques introduced show promise for further development, both to head towards a proof of the conjecture when the support has size 3 and for situations with larger support. We also show that if $y > (2x^2 + 2x + 1)/(x-2)$ 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 $(x,y) = (6,18)$.