Facet-Hamiltonicity

Abstract

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, Mütze, 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.

Figures (41)

• Figure 1: The dodecahedron with a Hamiltonian cycle ( a) and a facet-Hamil-tonian cycle ( b).
• Figure 2: A three-dimensional simple polytope that is not facet-Hamil-tonian.
• Figure 3: Facet-Hamiltonian cycles in a permutahedron ( a), and an associahedron ( c). Table ( b) shows the inductive structure of the facet-Hamil-tonian cycle on the 3-dimensional permutahedron in ( a). The cycle is obtained by combining two copies of the facet-Hamil-tonian path from $123$ to $312$ on the hexagon.
• Figure 4: A cylindrically closed rhombic strip encoding facet-Hamil-tonian cycles on the permutahedron for $n=5$.
• Figure 5: A rhombic strip of the 5 dimensional permutahedron and its corresponding Venn diagram. Note that the left and right side are identified as we are on the sphere.

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Conjecture 1
• proof
• Lemma 1
• proof