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 $k \geq 3$. 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 $Q$.

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