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Combinatorial generation via permutation languages. VII. Supersolvable hyperplane arrangements

par Brenner, Sofia;Cardinal, Jean ;McConville, Thomas;Merino, Arturo;Mütze, Torsten
Référence European journal of combinatorics, 135, 104367
Publication Publié, 2026-05

Article révisé par les pairs

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Résumé :

For an arrangement H of hyperplanes in Rn through the origin, a region is a connected subset of Rn∖H. The graph of regions G(H) has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from H. We aim to compute a Hamiltonian path or cycle in the graph G(H), i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if H is a supersolvable arrangement, then the graph of regions G(H) has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions P(H,R0), obtained by orienting the graph G(H) away from a particular base region R0. Our second main result is that if H is supersolvable and R0 is a canonical base region, then for any lattice congruence ≡ on P(H,R0)≕L, the cover graph of the quotient lattice L/≡ has a Hamiltonian path. These paths and cycles are constructed by a generalization of the well-known Steinhaus–Johnson–Trotter algorithm for listing permutations. This algorithm is a classical instance of a combinatorial Gray code , i.e., an algorithm for generating a set of combinatorial objects by applying a small change in each step. When applying our two main results to well-known supersolvable arrangements, such as the coordinate arrangement, and the braid arrangement and its subarrangements, we recover a number of known Gray code algorithms for listing various combinatorial objects, such as binary strings, binary trees, triangulations, rectangulations, acyclic orientations of graphs, congruence classes of quotients of the weak order on permutations, and of acyclic orientation lattices. These were obtained earlier from the framework of zigzag languages of permutations proposed by Hartung et al. (2022). When applying our main results to the type B Coxeter arrangement and its subarrangements, we obtain a number of new Gray code algorithms for listing (pattern-avoiding) signed permutations, symmetric triangulations, acyclic orientations of certain signed graphs, and in general for combinatorial families of Coxeter type B, which generalizes the theory of zigzag languages to signed permutations. Our approach also yields new Hamiltonicity results for large classes of polytopes, in particular signed graphic zonotopes and the type B quotientopes of Padrol et al. (2023).