Abstract
The saturation number of a graph F, written 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 et al. who determined sat(n,Kr) for all r≥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 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 sat(n,C3,C2k)=0 for all k≥3 and n≥2k+2. We extend this result to all odd cycles by proving that for any odd integer r≥5, sat(n,Cr,C2k)=0 for all 2k≥r+5 and n≥2kr.