Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3

Marco Buratti has conjectured that, given an odd prime p and a multiset L containing p - 1 integers taken from {1,..., p-1/2}, there exists a Hamiltonian path in the complete graph with p vertices whose multiset of edge-lengths is equal to L modulo p. We give a positive answer to this conjecture in the case of multisets of the type {1(a),2(b),3(c)} by completely classifying such multisets that are linearly or cyclically realizable.