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 \(\mathsf{P}(\mathcal{H}, R_0)\), 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 \(\mathsf{P}(\mathcal{H}, R_0) =: L\), the cover graph of the quotient lattice \(L/{\equiv}\) has a Hamiltonian path.

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order.