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 Bollobás, Saito, and Wormald.

我们在\（（2r + 1）\） -正则图中研究2 k个因子。汉森，Loten和托夫特证明，每一次\（（2R + 1）\） -regular图表至多2 - [R切刃具有2因子。我们通过证明\（k \ le（2r + 1）/ 3 \）来概括其结果，每个\（（2r + 1）\） -规则图最多具有\（2r-3（k-1）\）尖端的系数为2 k。对k的限制和对刀沿数量的限制都非常明显。我们表征具有完全\（2r-3（k-1）+1 \）前沿但没有2 k因子的图。对于\（k>（2r + 1）/ 3 \），根据Bollobás，Saito和Wormald的研究，有没有前沿的图没有2 k因子。