Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

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.