Perfect matchings in highly cyclically connected regular graphs

@article{Lukoka2017PerfectMI,
  title={Perfect matchings in highly cyclically connected regular graphs},
  author={Robert Lukoťka and Edita Rollov'a},
  journal={Journal of Graph Theory},
  year={2017},
  volume={100},
  pages={28 - 49}
}
A leaf matching operation on a graph consists of removing a vertex of degree 1 together with its neighbour from the graph. Let G be a d ‐regular cyclically ( d − 1 + 2 k ) ‐edge‐connected graph of even order, where k ≥ 0 and d ≥ 3 . We prove that for any given set X of d − 1 + k edges, there is no 1‐factor of G avoiding X if and only if either an isolated vertex can be obtained by a series of leaf matching operations in G − X , or G − X has an independent set that contains more than half of the… 

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