Ascend/descend algorithms on Cayley graphs

In a fundamental paper, F.P. Preparata and J.E. Vuillemin (1981) introduced the cube connected cycles graph and demonstrated a congestion free implementation of an ascend/descend algorithm. Subsequently, it was shown that the cube connected cycles graph is the Cayley graph of a group, the wreath product. We isolate the properties required of a Cayley graph that enable a congestion free implementation of an ascend/descend algorithm. We exhibit another family of graphs which we call supertoroids which possess this property, and we analyze the time complexity of the resulting implementation