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.
Highlights
Hypergraphs constitute a far-reaching generalization of graphs and a basic combinatorial construct but are notoriously difficult to work with
A perfect matching in a hypergraph is a collection of edges partitioning the vertex set
The obvious obstruction to containing a perfect matching is existence of an isolated vertex, and a natural guess is that this is the main obstruction
Summary
Hypergraphs constitute a far-reaching generalization of graphs and a basic combinatorial construct but are notoriously difficult to work with. A literal form of this assertion—the stopping time version—says that if we choose random edges sequentially, each uniform from those as yet unchosen, we w.h.p.1 have a perfect matching as soon as all vertices are covered This nice behavior does hold for graphs [3], but for hypergraphs remains conjectural (though at least the value it suggests for the threshold is correct). An interesting point here is that taking p large enough to avoid isolated vertices produces many more edges than other considerations—e.g., wanting a large expected number of perfect matchings—suggest This has been one motivation for the substantial body of work on models of random graphs in which isolated vertices are automatically avoided, notably random regular graphs (e.g., [22]) and the k-out model.
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