Representation of large matchings in bipartite graphs

TL;DR

This paper improves the upper bound on the minimum size of matchings needed in bipartite graphs to guarantee a full rainbow matching, reducing the previous asymptotic bound to a constant plus a linear term.

Contribution

It refines the upper bound for the function f(n), showing that f(n) is at most eil; rac{3}{2}n ceil + 1, thus advancing the understanding of rainbow matchings in bipartite graphs.

Findings

f(n) q eil; rac{3}{2}n ceil + 1 bound established

Improved the asymptotic bound from rac{3}{2}n + o(n) to a constant plus linear term

Progress towards conjecture f(n)=n+1 for large n

Abstract

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenm{\"u}ller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely $f(n) \le \lceil \frac{3}{2}n \rceil+1$.