Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture

Label the vertices of the complete graph Kv with the integers {0, 1, . . ., v − 1} and define the length of the edge between the vertices x and y to be min(|x−y|, v−|x−y|). Let L be a multiset of size v − 1 with underlying set contained in {1, . . ., ⌊v/2⌋}. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in Kv whose edge lengths are exactly L if and only if for any divisor d of v the number of multiples of d appearing in L is at most v − d. We introduce “growable realizations,” which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in {1, 4, 5} or in {1, 2, 3, 4} and a partial result when the underlying set has the form {1, x, 2x}. We believe that for any set U of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set U