Combinatorial generation via permutation languages. VII. Supersolvable hyperplane arrangements

TL;DR

This paper proves the existence of Hamiltonian cycles in the graph of regions for supersolvable hyperplane arrangements and Hamiltonian paths in certain quotient lattices, advancing combinatorial understanding of hyperplane arrangements.

Contribution

It establishes Hamiltonian cycles for the graph of regions in supersolvable arrangements and Hamiltonian paths in quotient lattices derived from these arrangements, extending previous combinatorial results.

Findings

Hamiltonian cycle exists in the graph of regions for supersolvable arrangements.

Hamiltonian path exists in quotient lattices of the poset of regions.

Results generalize to lattice quotients with canonical base regions.

Abstract

For an arrangement $\mathcal{H}$ of hyperplanes in $\mathbb{R}^n$ through the origin, a region is a connected subset of $\mathbb{R}^n\setminus\mathcal{H}$. The graph of regions $G(\mathcal{H})$ has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from $\mathcal{H}$. We aim to compute a Hamiltonian path or cycle in the graph $G(\mathcal{H})$, i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if $\mathcal{H}$ is a supersolvable arrangement, then the graph of regions $G(\mathcal{H})$ has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions $P(\mathcal{H},R_0)$, obtained by orienting the graph $G(\mathcal{H})$ away from a particular base region $R_0$. Our second main result is that if $\mathcal{H}$ is supersolvable and $R_0$ is a canonical base region, then for any lattice congruence $\equiv$ on $P(\mathcal{H},R_0)=:L$, the cover graph of the quotient lattice $L/\equiv$ has a Hamiltonian path. [...]