TITLE: Edge-transitive Cayley graphs of $S_n$
QUESTION [10 upvotes]: I came across the following question which I haven't seen before:
Question.  Fix $k\ge 3$.  For infinitely many $n$, does there exists a generating set $\langle R_n \rangle = S_n$, $|R_n|=k$, such that the corresponding (undirected) Cayley graph $\Gamma(S_n,R_n)$ is edge-transitive?  
Perhaps, there is a simple explicit combinatorial construction which would work for all large enough $n$ and some fixed $k$.  Are there any references I am missing?   
NOTE:  Now that I see a nice answer by Brendan McKay, let me mention the reason for the question.  I just learned the construction of edge-transitive expanders on $S_n$ for some sparse sequence of $n$, but my own simple construction to answer the question above worked only for $n=p+1$, where $p$ is a prime.

REPLY [10 votes]: Here's a partial answer.
Take the generators to be a set of equal length cycles that are disjoint except that they have one point in common.  For example $\langle (1,2,3,4), (1,5,6,7), (1,8,9,10)\rangle$.  I believe that will generate $S_n$ except where the cycles have odd length (in which case they will generate $A_n$). I didn't prove that but I tried a few cases and expect it is easy. Moreover, it is obvious that the Cayley graph will have automorphisms swapping the generators around.