论文标题
抽象多面体的平坦扩展
Flat extensions of abstract polytopes
论文作者
论文摘要
我们考虑构建抽象$（n+1）$ - polytope $ q $的问题，带有$ k $ facets同构与给定的$ n $ -polytope $ p $，其中$ k \ geq 3 $。特别是，我们考虑的情况是，我们希望$ q $为$（n-2，n）$ - 平坦，这意味着每个$（n-2）$ - 面对每个$ n $ face（facet）的事件。我们表明，如果$ p $承认给定$ k $的平面扩展名，那么$ p $的刻面图是$（k-1）$ - 可着色。相反，我们表明，如果Facet图为$（k-1）$ - 可着色和$ k-1 $是PRIME，则$ P $承认该$ K $的平面扩展名。我们还表明，如果$ p $是facet - 两部分，那么对于每个$ k $，都有一个平坦的扩展名$ p | k $，以便每种$ p $的自动形态都扩展到$ p | k $的自动形态。 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同构至$ q $。
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 $Q$.
