Exponentially many nowhere-zero Z3-, Z4-, and Z6-flows

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. Our results may be seen as a counting alternative to the well-known proofs of existence of Z3-, Z4-, and Z6-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen.