Rainbow Odd Cycles

@article{Aharoni2020RainbowOC,
  title={Rainbow Odd Cycles},
  author={Ron Aharoni and Joseph Briggs and Ron Holzman and Zilin Jiang},
  journal={SIAM J. Discret. Math.},
  year={2020},
  volume={35},
  pages={2293-2303}
}
We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n… 

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