Rainbow fractional matchings

TL;DR

This paper proves the existence of rainbow fractional matchings in hypergraphs under certain conditions, extending known results from bipartite graphs to hypergraphs using topological methods.

Contribution

It establishes a fractional version of the rainbow matching problem for hypergraphs, generalizing previous bipartite graph results with a novel topological proof.

Findings

Rainbow fractional matchings exist under specified conditions.

Number of sets needed reduces in r-partite hypergraphs when n is integer.

Topological methods are effective in hypergraph matching problems.

Abstract

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a set of edges from distinct $E_i$'s which supports such a fractional matching). When the hypergraph is $r$-partite and $n$ is an integer, the number of sets needed goes down from $rn$ to $rn-r+1$. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for $r$-partite hypergraphs with $r>2$. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.