• DOI:10.1007/s00373-022-02475-1
• Corpus ID: 232478758

A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings

@article{Lu2022ANO,
  title={A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings},
  author={Hongliang Lu and Xingxing Yu},
  journal={Graphs and Combinatorics},
  year={2022},
  volume={38},
  pages={1-8}
}

• Hongliang Lu, Xingxing Yu
• Published 1 April 2021
• Mathematics
• Graphs and Combinatorics

Rödl, Ruciński, and Szemerédi determined the minimum $$(k-1)$$ ( k - 1 ) -degree threshold for the existence of fractional perfect matchings in k -uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the d -degree threshold for the range $$k-1>d\ge k/2$$ k - 1 > d ≥ k / 2 . In this note, we prove the following exact degree threshold: let k ,  d be positive integers with $$k\ge 4$$ k ≥ 4 and $$k-1>d\ge k/2$$ k - 1 > d ≥ k / 2 , and let n be any integer with $$n\ge 2k(k-1)+1$$ n ≥ 2 k ( k - 1 ) + 1 . Then any n -vertex k -uniform hypergraph with minimum d -degree $$\delta _d(H)>{n-d\atopwithdelims ()k-d} -{n-d-(\lceil n/k\rceil -1)\atopwithdelims ()k-d}$$ δ d ( H ) > n - d k - d - n - d - ( ⌈ n / k ⌉ - 1 ) k - d contains a fractional perfect matching. This lower bound on the minimum d -degree is best possible. We also determine the minimum d -degree threshold for the existence of fractional matchings of size s , where $$0<s\le n/k$$ 0 < s ≤ n / k (when $$k/2\le d\le k-1$$ k / 2 ≤ d ≤ k - 1 ), or with s large enough and $$s\le n/k$$ s ≤ n / k (when $$2k/5<d<k/2$$ 2 k / 5 < d < k / 2 ).