Uniqueness of the extreme cases in theorems of Drisko and Erd\H{o}s-Ginzburg-Ziv

TL;DR

This paper characterizes the extremal cases in theorems by Drisko and Erdős-Ginzburg-Ziv, extending previous results to matchings of arbitrary sizes and providing new insights into rainbow matchings in bipartite graphs.

Contribution

It extends Drisko's theorem to matchings of arbitrary sizes and characterizes families of matchings that lack maximum rainbow matchings, also re-proves a key extremal case in Erdős-Ginzburg-Ziv theorem.

Findings

Characterization of families of 2n-2 matchings without a rainbow matching of size n.

Extension of Drisko's theorem to non-equal matching sizes.

Re-proof of the extremal case in Erdős-Ginzburg-Ziv theorem.

Abstract

Drisko \cite{drisko} proved (essentially) that every family of $2n-1$ matchings of size $n$ in a bipartite graph possesses a partial rainbow matching of size $n$. In \cite{bgs} this was generalized as follows: Any $\lfloor \frac{k+2}{k+1} n \rfloor -(k+1)$ matchings of size $n$ in a bipartite graph have a rainbow matching of size $n-k$. We extend this latter result to matchings of not necessarily equal cardinalities. Settling a conjecture of Drisko, we characterize those families of $2n-2$ matchings of size $n$ in a bipartite graph that do not possess a rainbow matching of size $n$. Combining this with an idea of Alon \cite{alon}, we re-prove a characterization of the extreme case in a well-known theorem of Erd\H{o}s-Ginzburg-Ziv in additive number theory.