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Aug 26, 2017 at 19:51 comment added Gerhard Paseman Of course, the graph could be a disjoint union of connected graphs each with a single cycle (If I ignore Joseph's comment above). I will say more after some hand computation. Gerhard "Not Ready For Foot Computation" Paseman, 2017.08.26.
Aug 26, 2017 at 19:23 comment added Joseph O'Rourke Actually, I have now verified that "there is only one cycle (of length $182$) in the transition graph for the space involving a path of length $6$ with a starting point of a permuted labeling of the variables."
Aug 26, 2017 at 19:20 history answered Gerhard Paseman CC BY-SA 3.0