The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) involutions, or (B) they are each other's inverses and of order >2. Note that


* non-isomorphic Cayley graphs can be isomorphic as graphs, so that the same cubic graph may arise as a Cayley graph of both types A and B;
* a single group can have non-isomorphic cubic Cayley graphs, so that the same group may have Cayley graphs of both types A and B;
* in particular, it might even be possible for a single group to give rise to non-isomorphic cubic Cayley graphs which are isomorphic as graphs.


What is known about the characterizations of, or relationship between, the two types A and B of cubic Cayley graphs, and in particular the situations in which they can coincide - either in the sense of being isomorphic as graphs, or of coming from the same group (or possibly both)?


I hope this question is not too vague.



Cubic Cayley (undirected) graphs
Everything that is not obviously forbidden can happen.


For example, a single group can have two Cayley graphs that are isomorphic (as graphs) but which are not of the same type. Such examples will definitely not be isomorphic as Cayley graphs (in your terminology).


I think if you take the dihedral group $D\_4=\langle a,b|a^4,b^2,(ba)^2\rangle$ then $\{a,a^{-1},b\}$ and $\{b,ba,ba^2\}$ yield the smallest example. (They are both isomorphic to the cube.)


Otherwise, your question IS a little vague but part of it is closely related to so-called Cayley isomorphic graphs or CI-graphs. There is a large literature on these, so maybe have a look to see if it helps.


