A note on exact minimum degree threshold for fractional perfect matchings
Abstract
Rödl, Ruciński, and Szemerédi determined the minimum $(k1)$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 $k1>d\ge k/2$. In this note, we prove the following exact degree threshold: Let $k,d$ be positive integers with $k\ge 4$ and $k1>d\geq k/2$, and let $n$ be any integer with $n\ge k^2$. Then any $n$vertex $k$uniform hypergraph with minimum $d$degree $\delta_d(H)>{nd\choose kd} {nd(\lceil n/k\rceil1)\choose kd}$ contains a fractional perfect matching. This lower bound on the minimum $d$degree is best possible. We also determine optimal minimum $d$degree conditions which guarantees the existence of fractional matchings of size $s$, where $0<s\le n/k$ (when $k/2\le d\le k1$), or with $s$ large enough and $s\le n/k$ (when $2k/5<d<k/2$).
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.00518
 Bibcode:
 2021arXiv210400518L
 Keywords:

 Mathematics  Combinatorics