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 \(2k/5<d<k/2\)).