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

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

Research output: Contribution to journalArticlepeer-review

## Abstract

We study 2k-factors in (2 r+ 1) -regular graphs. Hanson, Loten, and Toft proved that every (2 r+ 1) -regular graph with at most 2r cut-edges has a 2-factor. We generalize their result by proving for k≤ (2 r+ 1) / 3 that every (2 r+ 1) -regular graph with at most 2 r- 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 2 r- 3 (k- 1) + 1 cut-edges but no 2k-factor. For k> (2 r+ 1) / 3 , there are graphs without cut-edges that have no 2k-factor, as studied by Bollobás, Saito, and Wormald.

Original language English (US) 199-207 9 Graphs and Combinatorics 37 1 https://doi.org/10.1007/s00373-020-02242-0 Published - Jan 2021

## Keywords

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

## ASJC Scopus subject areas

• Theoretical Computer Science
• Discrete Mathematics and Combinatorics

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