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 $leftlfloor frac{v}{2}rightrfloor$ 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.
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