Label the vertices of the complete graph Kv with the integers {0, 1, . . . , v − 1} and define the length of the edge between 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, . . . , bv/2c}. 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 . MSC: 05C38, 05C78.
[1]
Alexander Rosa,et al.
On a Problem of Marco Buratti
,
2009,
Electron. J. Comb..
[2]
Anita Pasotti,et al.
On the Buratti-Horak-Rosa Conjecture about Hamiltonian Paths in Complete Graphs
,
2013,
Electron. J. Comb..
[3]
Marco Buratti,et al.
Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph
,
2013
.
[4]
Anita Pasotti,et al.
A new result on the problem of Buratti, Horak and Rosa
,
2014,
Discret. Math..
[5]
N. S. Barnett,et al.
Private communication
,
1969
.
[6]
Francesco Monopoli.
Absolute Differences Along Hamiltonian Paths
,
2015,
Electron. J. Comb..
[7]
Alberto Del Fra,et al.
Hamiltonian Paths in the Complete Graph with Edge-Lengths 1, 2, 3
,
2010,
Electron. J. Comb..
[8]
Anita Pasotti,et al.
New methods to attack the Buratti-Horak-Rosa conjecture
,
2021,
Discret. Math..