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, 2, 3} by completely classifying such multisets that are linearly or cyclically realizable.