$C_{2k}$-saturated graphs with no short odd cycles

The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erd\H{o}s, Hajnal, and Moon who determined $\textup{sat}(n,K_r)$ for all $r \geq 3$. Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman's generalized Tur\'an function, Kritschgau et.\ al.\ defined $\textup{sat}(n,H,F)$ to be the minimum number of copies of $H$ in an $n$-vertex $F$-saturated graph. They proved, among other things, that $\textup{sat}(n,C_3,C_{2k}) = 0$ for all $k \geq 3$ and $n \geq 2k +2$. We extend this result to all odd cycles by proving that for any odd integer $r \geq 5$, $\textup{sat}(n, C_r,C_{2k}) = 0$ for all $2k \geq r+5$ and $n \geq 2kr$.