  Label the vertices of the complete graph $K_v$ with the integers $\{ 0, 1,
\ldots, v-1 \}$ 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 $\{ 1, \ldots, \lfloor v/2 \rfloor \}$. The
Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in $K_v$
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 $U$.

Combinatorics