On the Aharoni-Berger conjecture for general graphs

Ron Aharoni, Eli Berger, Maria Chudnovsky, Shira Zerbib

Published 2020-12-30Version 1

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. There are quite a few asymptotic results, but one problem remains obstinate: prove that $n$ matchings of size $n$ have a rainbow matching of size $n-o(n)$. This is known when the graph if bipartite, and when the matchings are disjoint. 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 prove a result on rainbow paths, that if extended to alternating paths would prove the $n-o(n)$ conjecture.