New methods to attack the Buratti-Horak-Rosa conjecture

Abstract

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 $\lfloor \frac{v}{2}\rfloor$ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set $\{0,1,\dots ,v-1\}$ 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: $\{x,y,x+y\}$, $\{1,2,3,4\}$, $\{1,2,4,\dots ,2x\}$, $\{1,2,4,\dots ,2x,2x+1\}$. We also consider lists with many consecutive elements.

In this version we describe new methods that allowed us to improve the results of the previous "A new result on the problem of Buratti, Horak and Rosa"

References in corpus (2)

• On Partial Sums in Cyclic Groups
• A generalization of the problem of Mariusz Meszka