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≤(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.
- Graph factor
- Regular graph