# Flat extensions of abstract polytopes

### Abstract

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*.