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

TL;DR

This paper investigates the minimum number of certain subgraphs in graphs that are saturated with respect to even cycles, extending known results to all odd cycles and establishing conditions for zero subgraph counts.

Contribution

It generalizes previous results by proving that for any odd cycle, the saturation number with respect to even cycles is zero under specific conditions.

Findings

For any odd cycle $C_r$ with $r \\geq 5$, the saturation number with respect to $C_{2k}$ is zero.

The result holds for all $2k \\geq r+5$ and $n \\geq 2kr$.

Extends earlier work on triangles to all odd cycles in the context of saturated graphs.

Abstract

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 $n \geq 2kr$.