Rainbow Fractional Matchings
- 26 Downloads
We prove that any family E1,..., 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 Ei’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.
Mathematics Subject Classification (2010)05D15 55U10
Unable to display preview. Download preview PDF.
We are grateful to Dani Kotlar, Roy Meshulam and Ran Ziv for helpful discussions.
- R. Aharoni and E. Berger: Rainbow matchings in r-partite r-graphs, Electron. J. Combin. 16, Research Paper 119, (2009).Google Scholar