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 ≤ (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.

Regular Turán numbers of complete bipartite graphs

“…Much of the previous work on finding k-factors in graphs takes place in the setting where the host graph is regular. The main result of [22] allows the host graph to contain loops, and so applies in our setting. If we use this theorem, we do not know how many loops were contained in the k-factor, and so unfortunately we cannot complete the last step in the proof (adding a matching to the absolute points).…”

Largest 2-Regular Subgraphs in 3-Regular Graphs

“…. , 12}, the number of vertices is 3,5,7,11,15,25,31,59,71,113,127, respectively (see [3], for example).…”

“…For the general problem of minimizing f 2k (G) when G is (2r + 1)-regular, Kostochka et al [7] generalized [5] by showing that if k < (2r + 1)/3 and G has at most 2r − 3(k − 1) cut-edges, then G has a 2k-factor. Therefore, we are interested in how large a 2k-regular subgraph is guaranteed when there are more cut-edges.…”