On P completeness of some counting problems

We prove that the counting problems #1-in-3Sat, #Not-All-Equal 3Sat and #3-Colorability, whose decision counterparts have been the most frequently used in proving NP-hardness of new decision problems, are #P-complete. On one hand, the explicit #P-completeness proof of #1-in-3Sat could be useful to prove complexity results within unification theory. On the ither hand, the fact that #3-Colorability is #P-complete allows us to deduce immediately that the enumerative versions of a large class of NP-complete problems are #P-complete. Moreover, our proofs shed some new light on the interest of exhibiting linear reductions between NP problems