$$C_{2k}$$-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 et al. 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 Turan 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 $$n \ge 2kr$$ .