A Note on the Traveling Salesman Problem

It is shown that the “shortest” directed Hamiltonian tour in a graph G with metrically realizable distances has the property that between any vertex and its nearest neighbor in that tour there must be a vertex $v_0 $ from which the succeeding arc in the tour is no more than twice as long as the shortest arc from $v_0 $ in G Some related results and conjectures are described.