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I have recently been studying the paper Rainbow Cycles in Properly Edge-Colored Graphs (arXiv:2211.03291), which gives sufficient conditions for the existence of rainbow cycles in properly edge-colored graphs.

Motivated by this, I am interested in the enumeration of rainbow cycles in edge-colored complete graphs. Rainbow: every edges has different color. Specifically, let $k \leq \ell$ be positive integers. Suppose the complete graph $K_n$ is properly edge-colored with $\ell$ colors and $n$ is sufficiently large.

Question Do we know any upper or lower bounds for the number of rainbow cycles of length $k$ in such a coloring?

Thank you in advance for any pointers!

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