Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

TL;DR

This paper proves the existence of Hamiltonian cycles in the tope graphs of various classes of hyperplane arrangements, including simplicial, reflection, and supersolvable arrangements, with constructive methods.

Contribution

It extends known results by confirming Hamiltonicity in higher-dimensional and more complex arrangements, including all restrictions of finite reflection arrangements and supersolvable structures.

Findings

Hamiltonian cycles exist in all 3D simplicial arrangements in the Grünbaum–Cuntz catalogue.

All restrictions of finite reflection arrangements admit Hamiltonian cycles.

Supersolvable hyperplane arrangements and oriented matroids have Hamiltonian cycles, with a constructive proof.

Abstract

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Gr\"unbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all Weyl groupoids and crystallographic arrangements, admit Hamiltonian cycles. Finally, we further establish that all supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles, offering a constructive proof based on their inductive structure.