  We study $2k$-factors in $(2r+1)$-regular graphs. Hanson, Loten, and Toft
proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a
$2$-factor. We generalize their result by proving for $k\le(2r+1)/3$ that every
$(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor.
Both the restriction on $k$ and the restriction on the number of cut-edges are
sharp. We characterize the graphs that have exactly $2r-3(k-1)+1$ cut-edges but
no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have
no $2k$-factor, as studied by Bollob\'as, Saito, and Wormald.
