Before revision:  for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q \geq \alpha \Delta + 1 } ) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region.
After revision:  for example, we also obtain O(nn) mixing for q-colorings of triangle-free graphs of maximum degree \Delta when the number of colors satisfies q \geq \alpha \Delta + 1 } \lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region.