# Perfect fractional matchings in k-out hypergraphs

## TL;DR

This paper investigates conditions under which random k-out hypergraphs are likely to contain perfect fractional matchings, introducing new expansion concepts and extending results to r-uniform and r-partite hypergraphs.

## Contribution

It establishes the existence of a threshold k(r) for perfect fractional matchings in k-out hypergraphs and introduces a new hypergraph expansion notion.

## Key findings

- High probability of perfect fractional matchings in k-out hypergraphs for suitable k(r)
- Extension of results to r-uniform and r-partite hypergraphs
- New hypergraph expansion concept enabling these results

## Abstract

Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.