I have added that clarification.

Yes that is correct. I should have put that into the description.

Thanks for the fast response. I am thinking of the following: start with the identity $(1,2,\dots,n)$. Apply either generator 1 or generator 2 to get a new element. Proceed until all elements are covered. Bonus 1 means that after getting the last element, applying either generator 1 or 2 gets you the identity. I just added a "deterministic" requirement for the algorithm and by this I mean the sequences of generators to be used is known before hand.

I have completed the search: A total of 5,463,292,790,592 directed Hamiltonian cycles were found. I am currently trying to do as much check as I can to find any potential mistakes.

For those interested: It appears that there are > $2^{36}$ cycles and the search can easily be completed within 2 weeks. The main question happens to be shortage of storage space. – Ng Yong Hao 0 secs ago

Interesting. I guess I will try to find the error when I have the time then. As for the preprint, I am fine to wait for the actual paper to be available online. After all, I am quite interested to play around with the problem to see how far I can go. Thanks for your assistance again!

Also, I noticed that there is another paper on [Hamiltonian cycles in 6-cube][1] It appears that their result of 14754666508334433250560 Hamiltonian cycles (directed) is recorded in [OEIS A066037][2] It seems to me that this is the also the question that your paper is addressing. Perhaps there are some miscalculations for their values? My apologies if I made any mistakes here. [1]: arxiv.org/abs/1003.4391 [2]: oeis.org/A066037/internal

Many thanks for the information! From my experiments fixing 30/70 vertices reduces the brute force to instantaneous. With a number of symmetries I felt that $n=7$ should be doable. Nothing better than a confirmation though. Like what you mentioned, I found a lot of symmetries each only applicable to specific sequences. I am planning the pseudocode at the moment, actual program will probably take a while. Do you think the effort, mainly on the exploration of the symmetries, will be meaningful enough to be put into a paper? (seeing this is a rather specific case)

On a side note, I suppose answer acceptance process in MathOverflow is to select one as soon as you see a suitable candidate and switch as better ones appear? I cannot seem to locate the FAQ for this.

I feel that $n=7$ seems possible too. Although if its possible it looks like it will require some non trivial effort.