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Is there any MCMC mathod that allow me to uniformly sample from all feasible walks where the following restrictions apply:

1. visit vertex {1,2,3,..n} x1,x2,...,xn times
2. Doesn't go back in 1 step: i.e. no self loop
3. Doesn't go back in 2 steps: i.e. cannot have 1,2,3,2,....

Is following transposition chain irreducible?

randomly pick 2 vertices, swap them if possible.

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