Cut-Edges and Regular Factors in Regular Graphs of Odd Degree

Abstract

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 Bollobas, Saito, and Wormald.