### 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*, 2*x*}. 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*.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2659.be1

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications