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

TL;DR

This work introduces the Coprime Buratti-Horak-Rosa Conjecture, a coprime refinement of the BHR conjecture, and reduces the problem to the challenging case of supports of size $3$, specifically $L=\{1^a,x^b,y^c\}$ with $1<x<y$ and $\gcd(v,x)=\gcd(v,y)=1$. It develops grid-based linear realizations using omega-constructions and fauxset traversals, and introduces edge-replacement techniques (replacing 1-edges by $x$-edges, and later $x$-edges by $y$-edges) to cover broad parameter ranges, including explicit theorems that guarantee realizations when $a$ is large or when $y=tx$. The paper obtains both finite-regime and infinite-regime results: for instance, Coprime BHR holds for infinitely many $v$ when $y$ is sufficiently large relative to $x$, and it proves near-complete resolution for the fixed support ${1,6,18}$, leaving only a few exceptional $v$. Collectively, these constructions advance toward a full resolution of the Coprime BHR (and by extension the BHR) conjectures and yield substantial implications for known results on supports of size three and fixed-support instances.

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 $(x,y) = (6,18)$.

Figures (9)

• Figure 1: Standard linear realizations for $\{1^7, 8^{22} \}$ and $\{1^5, 5^{2}, 8^{21} \}$.
• Figure 2: The standard linear realization $\mathbf{h_2}$ for $\{1^6, 7^{18}\}$ and the standard linear realization $\mathbf{h_3}$ for $\{1^9, 9^{23}\}$.
• Figure 3: Standard linear realizations for $\{1^2, 3^4, 7^{15}\}$ and $\{1^3, 3^5, 9^{22}\}$.
• Figure 4: A standard linear realization for $\{1^5, 3^{10}, 16^{42} \}$ from the proof of Lemma \ref{['lem:fullfaux']}.
• Figure 5: A standard linear realization for $\{1^5, 3^{11}, 16^{42} \}$ from the proof of Lemma \ref{['lem:tailcurl']}.

Theorems & Definitions (56)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Example 1.6
• Lemma 1.7
• proof
• Theorem 1.8