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

Ollis, M. A., Pasotti, A., Pellegrini, M. A., Schmitt, J. R., Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture, <<ARS MATHEMATICA CONTEMPORANEA>>, 2022; 22 (4): N/A-N/A. [doi:10.26493/1855-3974.2659.be1] [http://hdl.handle.net/10807/214984]

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

#### 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
##### Scheda breve Scheda completa Scheda completa (DC)
2022
Inglese
Ollis, M. A., Pasotti, A., Pellegrini, M. A., Schmitt, J. R., Growable realizations: a powerful approach to the Buratti-Horak-Rosa Conjecture, <<ARS MATHEMATICA CONTEMPORANEA>>, 2022; 22 (4): N/A-N/A. [doi:10.26493/1855-3974.2659.be1] [http://hdl.handle.net/10807/214984]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10807/214984`
• ND
• 3
• 1