# 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}
}
• Published 19 July 2020
• Mathematics
• SIAM J. Discret. Math.
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 $K_{n… • Mathematics • 2019 For two graphs$G$and$H$, write$G \stackrel{\mathrm{rbw}}{\longrightarrow} H$if$G$has the property that every {\sl proper} colouring of its edges admits a {\sl rainbow} copy of$H\$. We study
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