Continuing the author’s earlier
investigation, this paper studies the behavior of paths on (convex) polyhedra relative
to the facets of the polyhedra. In Section 1, the polytopes which are polar to the
cyclic polytopes are shown to admit Hamiltonian circuits, and the fact that they do
leads to sharp upper bounds for the lengths of simple paths or simple circuits on
polyhedra of a given dimension having a given number of facets. Section 2 is
devoted to the conjecture, due jointly to Philip Wolfe and the author, that
any two vertices of a polytope can be joined by a path which never returns
to a facet from which it has earlier departed. This implies a well-known
conjecture of Warren Hirsch, asserting that n − d is an upper bound for
the diameter of d-dimensional polytopes having n facets. The Wolfe-Klee
conjecture is proved here for 3-dimensional polyhedra, and a stronger conjecture
(dealing with polyhedral cell-complexes) is established for certain special
cases.
