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. 



Edge-transitive Cayley graphs of $S_n$
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.


