We consider facet-Hamiltonian cycles of polytopes, defined as cycles in their skeleton such that every facet is visited exactly once. These cycles can be understood as optimal watchman routes that guard the facets of a polytope. We consider the existence of such cycles for a variety of polytopes, the facets of which have a natural combinatorial interpretation. In particular, we prove the following results: - Every permutahedron has a facet-Hamiltonian cycle. These cycles consist of circular sequences of permutations of $n$ elements, where two successive permutations differ by a single adjacent transposition, and such that every subset of $[n]$ appears as a prefix in a contiguous subsequence. With these cycles we associate what we call rhombic strips which encode interleaved Gray codes of the Boolean lattice, one Gray code for each rank. These rhombic strips correspond to simple Venn diagrams. - Every generalized associahedron has a facet-Hamiltonian cycle. This generalizes the so-called rainbow cycles of Felsner, Kleist, M\"utze, and Sering (SIDMA 2020) to associahedra of any finite type. We relate the constructions to the Conway-Coxeter friezes and the bipartite belts of finite type cluster algebras. - Graph associahedra of wheels, fans, and complete split graphs have facet-Hamiltonian cycles. For associahedra of complete bipartite graphs and caterpillars, we construct facet-Hamiltonian paths. The construction involves new insights on the combinatorics of graph tubings. We also consider the computational complexity of deciding whether a given polytope has a facet-Hamiltonian cycle and show that the problem is NP-complete, even when restricted to simple 3-dimensional polytopes.

翻译：我们研究多面体的面哈密顿环，其定义为多面体骨架中的环，使得每个面恰好被访问一次。这些环可理解为守卫多面体各面的最优巡逻路线。我们考察了各类多面体是否存在此类环，这些多面体的面具有自然的组合解释。具体而言，我们证明了以下结果：- 每个排列多面体都存在面哈密顿环。这些环由 $n$ 个元素的排列循环序列构成，其中相邻排列仅相差一个相邻对换，且每个 $[n]$ 的子集都作为连续子序列的前缀出现。我们将这些环与称为菱形带的结构相关联，这些菱形带编码了布尔格的分层格雷码，每个秩对应一个格雷码。这些菱形带对应于简单的维恩图。- 每个广义关联多面体都存在面哈密顿环。这将Felsner、Kleist、Mütze和Sering（SIDMA 2020）提出的所谓彩虹环推广到任意有限型关联多面体。我们将这些构造与Conway-Coxeter饰带及有限型簇代数的二分带联系起来。- 轮图、扇图和完全分裂图的图关联多面体具有面哈密顿环。对于完全二分图和毛虫图的关联多面体，我们构造了面哈密顿路径。该构造涉及图管道组合结构的新见解。我们还研究了判定给定多面体是否存在面哈密顿环的计算复杂度问题，证明该问题属于NP完全问题，即使限制在简单三维多面体上亦然。