  This paper deals with lattice congruences of the weak order on the symmetric
group, and initiates the investigation of the cover graphs of the corresponding
lattice quotients. These graphs also arise as the skeleta of the so-called
quotientopes, a family of polytopes recently introduced by Pilaud and Santos
[Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra,
associahedra, hypercubes and several other polytopes. We prove that all of
these graphs have a Hamilton path, which can be computed by a simple greedy
algorithm. This is an application of our framework for exhaustively generating
various classes of combinatorial objects by encoding them as permutations. We
also characterize which of these graphs are vertex-transitive or regular via
their arc diagrams, give corresponding precise and asymptotic counting results,
and we determine their minimum and maximum degrees. Moreover, we investigate
the relation between lattice congruences of the weak order and pattern-avoiding
permutations.
