Large matchings in bipartite graphs have a rainbow matching

TL;DR

This paper improves the upper bound on the minimum size of matchings needed in bipartite graphs to guarantee a rainbow matching, advancing the understanding of a conjecture related to combinatorial matchings.

Contribution

The paper proves a tighter upper bound of 5/3 n for the function g(n), improving previous bounds and contributing to the conjecture on rainbow matchings in bipartite graphs.

Findings

Proved that g(n) a 5/3 n upper bound

Improved previous bound of a  7/4 n

Contributes to longstanding conjecture in combinatorics

Abstract

Let $g(n)$ be the least number such that every collection of $n$ matchings, each of size at least $g(n)$, in a bipartite graph, has a full rainbow matching. Aharoni and Berger \cite{AhBer} conjectured that $g(n)=n+1$ for every $n>1$. This generalizes famous conjectures of Ryser, Brualdi and Stein. Recently, Aharoni, Charbit and Howard \cite{ACH} proved that $g(n)\le\lfloor\frac{7}{4}n\rfloor$. We prove that $g(n)\le\lfloor\frac{5}{3} n\rfloor$.