An improved bound on the sizes of matchings guaranteeing a rainbow matching

Abstract

A conjecture by Aharoni and Berger states that every family of n matchings of size n + 1 in a bipartite multigraph contains a rainbow matching of size n. In this paper we prove that matching sizes of (3/2 + o(1))n suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best-known one in the case where we only aim to find a rainbow matching of size n – 1. This improves previous results by Aharoni, Charbit and Howard, and Kotlar and Ziv.