On the structure of the monotone asymmetric travelling salesman polytope I: hypohamiltonian facets

The monotone asymmetric travelling salesman polytope P?ⁿ_T is defined to be the convex hull of the incidence vectors of all hamiltonian circuits and all subsets of these in a complete diagraph of order n. We prove that certain hypohamiltonian diagraphs G=(V,E), i.e. diagraphs which are not hamiltonian but such that G–υ is hamiltonian for all υ?V, induce facets x(E)?n–1 of P?ⁿ_T. This result indicates that P?ⁿ_T has very complicated facets and that it is very unlikely that an explicit complete characterization of P?ⁿ_T can ever be given.