Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3 (Q2380478)

Summary: Marco Buratti has conjectured that, given an odd prime \(p\) and a multiset \(L\) containing \(p-1\) integers taken from \(\{1,\dots,{p-1\over 2}\}\), 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.