Buratti–Horak–Rosa conjecture

Establish that a multiset L of size v−1 with support contained in {1,2,…,⌊v/2⌋} has a cyclic realization if and only if, for every divisor d of v, the number of elements of L divisible by d is at most v−d.

Background

The Buratti–Horak–Rosa (BHR) Conjecture extends Buratti’s conjecture from prime-order complete graphs to arbitrary v. It characterizes cyclic realizability through a necessary divisor condition: no divisor d of v may divide more than v−d entries of the multiset.

The paper studies linear realizations as a construction tool for cyclic realizations and focuses on the unresolved support-size-three cases of this conjecture.

References

Conjecture 1.1. 9 Let v be prime. If L is a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}, 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 1.1, Section 1 (Introduction)

Conjecture 1.2. 5, 8 Let L be a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}. Then L is cyclically realizable if and only if for any divisor d of v the number of multiples of d in L is at most v − d.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.2, Section 1 (Introduction)

Conjecture 1.2. 5, 8 Let L be a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}. Then L is cyclically realizable if and only if for any divisor d of v the number of multiples of d in L is at most v − d.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.2, Section 1 (Introduction)

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)

Conjecture 1.2. 5, 8 Let L be a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}. Then L is cyclically realizable if and only if for any divisor d of v the number of multiples of d in L is at most v − d.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.2, Section 1 (Introduction)

Conjecture 1.1. 9 Let v be prime. If L is a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}, 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 1.1, Section 1 (Introduction)

Conjecture 1.2. 5, 8 Let L be a multiset of size v − 1 with supp(L) ⊆ {1, 2, . . . , ⌊v/2⌋}. Then L is cyclically realizable if and only if for any divisor d of v the number of multiples of d in L is at most v − d.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.2, Section 1 (Introduction)