Listing faces of polytopes

TL;DR

The paper develops a comprehensive framework to list all faces of combinatorial polytopes by constructing Hamiltonian cycles in the face-lattice cover graphs $G(L(P))$ and, in particular, facet-Hamiltonian cycles in truncated polytopes. Central to the approach are rhombic strips, which induce Gray-code like orderings of faces and yield loopless algorithms for listing faces, ordered set partitions, and polygon dissections. The authors prove Hamiltonicity for a wide range of polytopes (hypercubes, permutahedra and their variants, associahedra, cyclic polytopes, 3D polytopes, graph-associahedra of chordal graphs, and quotientopes) and resolve the facet-Hamiltonian conjecture for $B$-permutahedra via rhombic-strip constructions and truncation arguments. The work also provides constructive, time- and space-efficient algorithms for generating these face-lattice traversals and corresponding combinatorial objects, with several concrete implementations cited. Overall, the results establish a broad, algorithmically effective theory for enumerating faces and related structures across a wide spectrum of combinatorial polytopes, deepening connections between Hamiltonicity, rhombic-strip structures, and Gray-code generation.

Abstract

This paper investigates the problem of listing faces of combinatorial polytopes, 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 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.

Figures (37)

• Figure 1: Examples of 3-dimensional combinatorial polytopes: (a) the hypercube; (b) the permutahedron; (c) the associahedron.
• Figure 2: (a) A 3-polytope $P$; (b) a Schlegel diagram of $P$, a 3-connected plane graph; (c) the face lattice of $P$; (d1)+(d2) a Hamiltonian cycle in $G(L(P))$; (e1)+(e2) two distinct facet-Hamiltonian cycles of $P$ (of different lengths); (f) a rhombic strip of $G(L(P))$. The rhombi are colored according to ranks for clarity. In this and all of the following pictures of rhombic strips in our paper, we display a grayed-out copy of the leftmost vertices at the right-hand side, in order to depict the 'wrap-around' edges on the sphere.
• Figure 3: Hamiltonian cycles in the cover graph of the face lattice of hypercubes $Q_n$ for $n=1,2,3,4$ (a1)--(a4) and permutahedra $\Pi_n$ for $n=2,3,4$ (b2)--(b4). The encoding for hypercubes is 0=white, 1=black, [1.0]- =gray and for permutahedra it is 1=blue, 2=green, 3=yellow, 4=red, where for values in the same block of an ordered set partition the corresponding rectangle is striped horizontally with the colors of values in that block.
• Figure 4: Hamiltonian cycles in the cover graph of the face lattice of the associahedron $A_n$ for $n=4,5,6$. The vertical bars delimit groups of dissections obtained from the same parent dissection in the previous sequence.
• Figure 5: The two partition classes of the bipartite graph $G(L(P))$.

Theorems & Definitions (48)

• Conjecture 1
• Conjecture 2: MR4863586
• Theorem 3: Folklore
• Theorem 4: Folklore
• proof : Proof 1 of Theorem \ref{['thm:Q-RS']}
• proof : Proof 2 of Theorem \ref{['thm:Q-RS']}
• Theorem 5
• proof
• Theorem 6
• proof