New methods to attack the Buratti-Horak-Rosa conjecture

• Published in: Discrete mathematics Vol. 344; no. 9; p. 112486
• Main Authors: Ollis, M.A., Pasotti, Anita, Pellegrini, Marco A., Schmitt, John R.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2021
• Subjects: Complete graph; Edge-length; Graceful permutation; Hamiltonian path; Linear realization; Linear realization; Edge-length; Complete graph; Hamiltonian path; Graceful permutation
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2021.112486

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list L of v−1 positive integers not exceeding ⌊v2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,…,v−1} if and only if, for every divisor d of v, the number of multiples of d appearing in L is at most v−d. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: {x,y,x+y}, {1,2,3,4}, {1,2,4,…,2x}, {1,2,4,…,2x,2x+1}. We also consider lists with many consecutive elements.