A Hamiltonian circuit solution has been discovered for the Rubik’s Cube, allowing it to pass through all positions without repeating any. The solution involves a sequence of quarter-turn moves and utilizes nested subgroups to connect cosets and create larger cycles. Surprisingly, the entire circuit consists of only five face layers, with no use of the back layer. The unique methodology of connecting cosets in groups of three and utilizing specific move sequences allows for the construction of the circuit. The detailed construction process, including specifications and files, can be found on the cuBer Bruce home page. (Note: The content assumes knowledge of group theory and Rubik’s Cube conventions)
https://bruce.cubing.net/ham333/rubikhamiltonexplanation.html