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We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the socalled uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q geq 3 and Delta geq 3, we develop algorithms that produce samples within error o(1) from the q -state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilizing the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p, q on random Delta-regular graphs for all values of q geq 1 and p < pc (q, Delta), where pc (q, Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs. [ABSTRACT FROM AUTHOR]