Adjacency of the Traveling Salesman Tours and $0 - 1$ Vertices

A necessary and sufficient condition is given for adjacency on the convex hull of 0–1 feasible points. The class of problems for which this condition is valid includes the set partitioning problem. A strong bound is derived for the diameter of the polytope associated with the convex hull of $0 - 1$ feasible points. A counterexample is given to a published necessary and sufficient condition for nonadjacency of two traveling salesman tours on their convex hull. A necessary condition is obtained for two tours to be nonadjacent on their convex hull. A sufficient condition for nonadjacency is also given. Examples are provided for the traveling salesman problem to show that neither the necessary condition is sufficient nor the sufficient condition is necessary. Finally, some adjacency properties are given for the traveling salesman tours on the assignment polytope.