Cut-edges and regular factors in regular graphs of odd degree

Alexander V. Kostochka, André Raspaud, Bjarne Toft, Douglas B. West*, Dara Zirlin

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


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.
Original languageEnglish
JournalGraphs and Combinatorics
Pages (from-to)199–207
Publication statusPublished - 2021


  • Cut-edge
  • Graph factor
  • Matching
  • Regular graph


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