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

Combinatorics