On unique graph 3-colorability and parsimonious reductions in the plane

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. A consequence of our result is the DP-completeness of Unique 3-Colorability and Unique Planar 3-Colorability under random P-time reductions. It also gives a finer and unified proof of the #P-completeness of #3-Colorability that was first obtained by Linial for the general case, and later by Hunt et al. for the planar case. Previous authors’ reductions were either weakly parsimonious with a multiplication of the numbers of solutions by an exponential factor, or involved #P-complete intermediate counting problems derived from trivial “yes”-decision problems.