Generating All Permutations Without Repetition Using Two Generators It is well known that all symmetric group can be generated using two generators
The two generators are:
1) $(1,2)$
2) $(1,2,3,\dots ,n)$

Question: Is there a deterministic algorithm to generate all permutations without repetition using only these two generators?
  (Bonus 1: The algorithm generates the permutations in a cycle. Bonus 2: Not requiring the inverse of generator 2)
  Edit: As point out by John, this is equivalent to a Hamiltonian path in the Cayley graph of $S_n$ with these two generators.

It is easy to generate all of them without repetition using $n-1$ generators, by the
Steinhaus-Johnson-Trotter algorithm.
It is easy to generate all of them, with repetition, using two generators.
However I was unable to find a way to generate all without repetition and using only two generators.
As this approach seems natural, I suspect someone should have worked on it but I was unable to find any references online.  
Does anyone knows the status of this problem?
 A: There has been a recent paper by Sawada and Williams where the problem is solved for the harder variant where you always shift in the same direction, i.e., you never require the inverse of generator 2 (Bonus 2 problem).
The paper is available here:
https://epubs.siam.org/doi/abs/10.1137/1.9781611975031.37
A demonstration of that algorithm can be run on the Combinatorial Object Server website:
http://page.math.tu-berlin.de/~muetze/cos/
Go to "Permutations", then select "Prefix swaps and rotations (Sawada-Williams)".
A: Your question is a special case of the Lovász conjecture, which says that all Cayley graphs are Hamiltonian.  According to Igor Pak (Hamiltonian Paths in Cayley Graphs) there is an explicit Hamiltonian Cycle on your two generators which requires only linear space to compute.  The construction is purported to reside in this paper (which I cannot access):
R. C. Compton, S. G. Williamson, Doubly adjacent Gray codes for the symmetric group,
Linear and Multilinear Algebra 35 (1993), 237–293.
