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

M. A. Ollis, Anita Pasotti, Marco A. Pellegrini, John R. Schmitt

Abstract


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.


Keywords


Hamiltonian path, complete graph, edge-length, growable realization

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DOI: 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