  The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching,
which can be used to transform edge-colorings, is here considered for
3-edge-colorings of cubic graphs. Computational results for edge-Kempe
switching of cubic graphs up to order 30 and bipartite cubic graphs up to order
36 are tabulated. Families of cubic graphs of orders $4n+2$ and $4n+4$ with
$2^n$ edge-Kempe equivalence classes are presented; it is conjectured that
there are no cubic graphs with more edge-Kempe equivalence classes. New
families of nonplanar bipartite cubic graphs with exactly one edge-Kempe
equivalence class are also obtained. Edge-Kempe switching is further connected
to cycle switching of Steiner triple systems, for which an improvement of the
established classification algorithm is presented.
