2024/12/07 by Onur Ağırseven, Agirseven, Onur, M. A. Ollis +1
Computer Science · Mathematics · #05C38 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2412.05750
openalex publication_date 2024/12/07 · openalex created_date 2024/12/12 · openalex updated_date 2026/07/28
We propose a "Coprime Buratti-Horak-Rosa (BHR) Conjecture": If L is a multiset of size v-1 with support contained in \1, 2, …, \lfloor v/2 \rfloor\ such that gcd(v,x) = 1 for all x ∈ 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 = \1a, xb, yc\, where 1 2x-2, and b ≥ y-2x+2; a ≥ x for y=tx, with x and t odd, and b ≥ tx+2t-3; a ≥ 7 for x=3 and b ≥ 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 > (2x2 + 2x + 1)/(x-2) then the Coprime BHR Conjecture holds for \1a,xb,yc\ 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).