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arXiv · 2610.10424

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

Hyunwoo Lee
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Abstract

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 $\left((1 + o_r(1))\frac{r}{e}\right)^n$ by a factor of $2^{(1 + o(1))n}$ and is asymptotically tight for large $r$. As a direct consequence, we determine asymptotically tight bounds on the number of $2$-factorizations of a given $2r$-regular simple graph for every sufficiently large $r$.

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Hyunwoo Lee. 2026-10-07. A counting version of Petersen's $2$-factor theorem. https://arxiv.org/abs/2610.10424

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