Rainbow Paths and Large Rainbow Matchings

  • Ron Aharoni
  • Eli Berger
  • Maria Chudnovsky
  • Shira Zerbib


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. 

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