Listing faces of polytopes

Abstract

This paper investigates the problem of listing faces of polytopes that represent combinatorial objects, such as hypercubes, permutahedra, associahedra, and their generalizations. Firstly, we consider the face lattice, which is the inclusion order of all faces of a polytope, and we seek a Hamiltonian cycle in its cover graph, i.e., for any two consecutive faces, one must be a subface of the other, and their dimensions differ by $1$. We construct such Hamiltonian cycles for hypercubes, permutahedra, $B$-permutahedra, associahedra, cyclic polytopes, 3-dimensional polytopes, graph associahedra of chordal graphs, and quotientopes. Secondly, we consider facet-Hamiltonian cycles, which are cycles on the (1-)skeleton of a polytope that enter and leave every facet exactly once. This notion was recently introduced by Akitaya, Cardinal, Felsner, Kleist, and Lauff [SODA 2025], where the authors conjectured that $B$-permutahedra admit a facet-Hamiltonian cycle for all dimensions. We construct such facet-Hamiltonian cycles in this paper, thus establishing their conjecture as a theorem. A key tool we use are so-called rhombic strips, which are planar spanning subgraphs of the cover graph of the face lattice in which every face is a 4-cycle. Specifically, we construct a rhombic strip in the face lattice of the hypercube of any dimension, and characterize the existence of rhombic strips in the face lattice of 3-dimensional polytopes. Our constructions yield time- and space-efficient algorithms for computing the aforementioned cycles and thus for listing the corresponding combinatorial objects, including ordered set partitions and dissections of a convex polygon.