Flat extensions of abstract polytopes
We consider the problem of constructing an abstract (n+1)-polytope Q with k facets isomorphic to a given n-polytope P, where k ≥ 3. In particular, we consider the case where we want Q to be (n−2, n)-flat, meaning that every (n−2)-face is incident to every n-face (facet). We show that if P admits such a flat extension for a given k, then the facet graph of P is (k−1)-colorable. Conversely, we show that if the facet graph is (k−1)-colorable and k−1 is prime, then P admits a flat extension for that k. We also show that if P is facet-bipartite, then for every even k, there is a flat extension P|k such that every automorphism of P extends to an automorphism of P|k. Finally, if P is a facet-bipartite n-polytope and Q is a vertex-bipartite m-polytope, we describe a flat amalgamation of P and Q, an (m+n−1)-polytope that is (n−2, n)-flat, with n-faces isomorphic to P and co-(n−2)-faces isomorphic to Q.