Complexity of unique coloring of graphs
This paper by Régis Barbanchon might be of interest. From the abstract:

We prove that the Satisfiability (resp. planar Satisfiability) problem
is parsimoniously P-time reducible to the 3-Colorability (resp. Planar
3-Colorability) problem, that means that the exact number of solutions
is preserved by the reduction, provided that 3-colorings are counted
modulo their six trivial color permutations. In particular, the
uniqueness of solutions is preserved, which implies that Unique
3-Colorability is exactly as hard as Unique Satisfiability in the
general case as well as in the planar case...