A counting version of Petersen's $2$-factor theorem

A classical result of Petersen states that every regular graph of even degree has a $2$-factor. We prove that every $n$-vertex $2r$-regular simple graph contains at least $\left((1 + o_r(1))\frac{2r}{e}\right)^n$ distinct $2$-factors. This improves the previously known lower bound $\left((1 + o_r(1)