Rainbow cycles vs. rainbow paths

An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Tur\'an number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Tur\'an numbers was introduced by Keevash, Mubayi, Sudakov, and Verstra\"ete. Johnson and Rombach introduced the following rainbow-version of generalized Tur\'an problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored graph with no rainbow copy of $F$. In this paper we investigate the case $\mathrm{ex}^*(n,C_\ell,P_\ell)$ and give a general upper bound as well as exact results for $\ell = 3,4,5$. Along the way we establish a new best upper bound on $\mathrm{ex}^*(n,P_5)$. Our main motivation comes from an attempt to improve bounds on $\mathrm{ex}^*(n,P_\ell)$, which has been the subject of several recent manuscripts.