Interchange graphs and the Hamiltonian cycle polytope

Abstract

This paper answers the (non)adjacency question for the whole spectrum of Hamiltonian cycles on the Hamiltonian cycle polytope (HC-polytope), also called the symmetric traveling salesman polytope, namely from Hamiltonian cycles that differ in only two edges through Hamiltonian cycles that are edge disjoint. The HC-polytope is the convex hull of the characteristic vectors corresponding to the Hamiltonian cycles of K n (n ⩾ 3). Let2 ⩽ k ⩾ n. The k -interchange graph is the graph with as vertices the1/2(n − 1)! Hamiltonian cycles of K n, and an edge between two vertices if and only if the corresponding Hamiltonian cycles differ in an interchange of k edges. It is shown that the 2- and the 3-interchange graphs are the only ones that are subgraphs of the skeleton of the HC-polytope; the ( n − 1- and the n-interchange graphs are the only ones that do not have edges in common with the skeleton. For each k with4 ⩽ k ⩽ n − 2, there are Hamiltonian cycles that are adjacent and cycles that are nonadjacent on the skeleton. Finally, the Hamiltonicity of k-interchange graphs is solved for several values of k.