Rainbow matchings and rainbow connectedness

TL;DR

This paper proves the Aharoni-Berger Conjecture for bipartite graphs with matchings of size close to n, using novel connectedness concepts in colored directed graphs, advancing understanding of rainbow matchings.

Contribution

It establishes the conjecture for matchings of size n + o(n) when they are edge-disjoint, and offers an alternative proof for larger matchings of size at least φn + o(n), introducing new connectedness notions.

Findings

Proves the conjecture for matchings of size n + o(n) with edge-disjointness.

Provides an alternative proof for matchings of size at least φn + o(n).

Introduces a new concept of connectedness in colored directed graphs.

Abstract

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. In the case when the matchings are much larger than n + 1, the best bound is currently due to Clemens and Ehrenm\"uller who proved the conjecture when the matchings are of size at least 3n/2 + o(n). When the matchings are all edge-disjoint and perfect, then the best result follows from a theorem of H\"aggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least $\phi n + o(n)$ where $\phi\approx 1.618$ is the Golden Ratio. Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.