Geometric shellings of 3-polytopes

A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3-polytopes. Recently, a graph theoretical characterization of geometric shellings of 3-polytopes were given by Holt & Klee and Mihalisin & Klee. We rst give a characterization of shellings of 3-polytopes. Then we show su cient conditions for a shellings to be geometric: the rst and the last facet being adjacent, any facet (except the rst two) being adjacent to no less than two previous facets or the induced orders being geometric shellings for two smaller polytopes made by dividing the polytope at a triple of facets adjacent to each other but not sharing a vertex. Simple 3-polytopes allow perturbations of facets, thus may have more chance a shelling is geometric. As su cient conditions for this case we show: the induced order being a geometric shelling for a smaller polytope made by removing a triangular or a quadrilateral facet or joining two consecutive facets in a shelling or the polytope only having triangular or quadrilateral facets. A nongeometric shelling of a (simplicial) 3-polytope was rst shown by Smilansky. We show such example for a simple 3-polytope, which is minimal with respect to the number of facets. The discussions proceed in the polar setting: as total orders of vertices of the polar polytope. All of our main results can be stated in graph theoretical terms. The proof techniques used are elementary topology, graph theory, network ows and local changes of polytopes.