Coprime BHR conjecture

Prove that every multiset L of size v−1 whose support is contained in {1,2,…,⌊v/2⌋} and whose every element is coprime to v has a cyclic realization.

Background

The Coprime BHR Conjecture considers the instances of the BHR Conjecture for which the divisor condition is vacuous because every supported length is coprime to v. It is intermediate between Buratti’s conjecture and the full BHR Conjecture.

The paper uses equivalence under automorphisms of the cyclic group to normalize one support element to 1 and investigates support-size-three families, including supports of the form {1,y−k,y}.

References

Conjecture 1.3. 2 If L is a multiset of size v − 1 such that supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋} and gcd(v, x) = 1 for all x ∈ L, then L is cyclically realizable.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.3, Section 1 (Introduction)