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

Abstract

Label the vertices of the complete graph K v 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 K v 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 .

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

“…In this section we give the necessary definitions and constructions that we use. Proofs of their correctness may be found in [7].…”

“…Let L = {2, 4 8 , 5 3 } (where exponents indicate multiplicity in a multiset). The Hamiltonian path [6,10,5,1,9,11,2,7,3,12,8,4,0] in K 13 has edge-lengths 4,5,4,5,2,4,5,4,4,4,4,4 and so realizes L.…”

“…Two early papers independently covered those of size at most 2 [3,4], followed shortly afterwards by a proof for {1, 2, 3}, the first three element set to be solved [2]. Subsequent work covers subsets of {1, 2, 3, 4} [7] or {1, 2, 3, 5} [9], {1, 2, 6} and {1, 2, 8} [8], and {1, 4, 5} [7]. In addition to this, there are strong partial results for a wide variety of underlying sets.…”

“…In addition to this, there are strong partial results for a wide variety of underlying sets. See [7,Theorem 1.2] for a summary of the position as of mid-2021. We shall use one of these partial results in Sections 4 and 5: when x is even, the multiset {1 a , 2 b , x c } is realizable if a + b ≥ x − 1 [8].…”

“…In the next section we introduce the notion of a growable realization, as developed in [7], which is our primary tool, and prove some new technical results providing constraints on when they can exist. For a given underlying set U , the approach divides the work into π(U ) = x∈U x cases.…”