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

TL;DR

This paper improves the theoretical bound on the size of matchings needed to guarantee a rainbow matching in bipartite multigraphs, approaching the conjectured optimal bound.

Contribution

It proves that matchings of size approximately 1.5 times n suffice to ensure a rainbow matching of size n, advancing the understanding of the problem.

Findings

Established an asymptotic bound of (3/2 + o(1)) n for rainbow matchings.

Improved upon previous bounds by Aharoni, Charbit, Howard, Kotlar, and Ziv.

Bridged the gap towards the conjectured optimal bound.

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 case 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.