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

Abstract

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