An approximate version of a conjecture of Aharoni and Berger

TL;DR

The paper proves an approximate version of a conjecture by Aharoni and Berger, showing that under certain conditions on edge counts in bipartite multigraphs, a rainbow matching covering all colours exists.

Contribution

It establishes that if each colour class has at least n + o(n) edges, then a rainbow matching using all colours is guaranteed, advancing understanding of the conjecture.

Findings

Proves an approximate version of the Aharoni-Berger Conjecture.

Shows existence of rainbow matchings with nearly optimal edge counts.

Links the problem to longstanding questions about Latin squares.

Abstract

Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by $n$ colours with at least $n+1$ edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the Aharoni-Berger Conjecture is proved---it is shown that if there are at least $n+o(n)$ edges of each colour in a proper $n$-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour.