Rainbow fractional matchings [article]

Ron Aharoni, Ron Holzman, Zilin Jiang
2019 arXiv   pre-print
We prove that any family E_1, ... , E_ rn 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
more » ... s, 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.
arXiv:1805.09732v2 fatcat:lezlsdhkljhafl2d7sg3tvxbku