Rainbow paths and large rainbow matchings

TL;DR

This paper improves lower bounds on the size of rainbow matchings in graphs and hypergraphs, and discusses related conjectures on rainbow paths, providing new theoretical results in combinatorics.

Contribution

It establishes a new lower bound of rac{2}{3}n - 1or rainbow matchings in graphs, surpassing the trivial bound, and extends results to hypergraphs and special graph classes.

Findings

Proved a lower bound of rac{2}{3}n - 1or rainbow matchings.

Extended bounds to hypergraphs and specific graph classes.

Confirmed the non-alternating path conjecture.

Abstract

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for hypergraphs. For $\{C_3,C_5\}$-free graphs and for disjoint matchings we obtain a lower bound of $\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture.