A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings

Abstract

Rödl, Ruciński, and Szemerédi determined the minimum $(k-1)$-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 $k-1>d\ge k/2$. In this note, we prove the following exact degree threshold: let k, d be positive integers with $k\ge 4$ and $k-1>d\ge k/2$, and let n be any integer with $n\ge 2k(k-1)+1$. Then any n-vertex k-uniform hypergraph with minimum d-degree $\delta _d(H)>{n-d\atopwithdelims ()k-d} -{n-d-(\lceil n/k\rceil -1)\atopwithdelims ()k-d}$ 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 $0<s\le n/k$ (when $k/2\le d\le k-1$), or with s large enough and $s\le n/k$ (when $2k/5<d<k/2$).