On saturation numbers of complete multipartite graphs and even cycles

TL;DR

This paper investigates the asymptotic behavior of the saturation number for complete multipartite graphs and large even cycles, extending previous results and partially resolving a longstanding conjecture in extremal graph theory.

Contribution

It determines the asymptotic saturation numbers for these classes of graphs, extending prior work and addressing an open conjecture.

Findings

Established asymptotic formulas for $ ext{sat}(n, F)$ for specified graphs.

Extended previous results by Bohman, Fonoberova, and Pikhurko.

Partially resolved a conjecture of F"uredi and Kim.

Abstract

Given positive integer $n$ and graph $F$, the saturation number $\mathrm{sat}(n, F)$ is the minimum number of edges in an edge-maximal $F$-free graph on $n$ vertices. In this paper, we determine asymptotic behavior of $\mathrm{sat}(n, F)$ when $F$ is either a complete multipartite graph or a cycle graph whose length is even and large enough. This extends a result by Bohman, Fonoberova, and Pikhurko from 2010 as well as partially resolves a conjecture of F\"uredi and Kim from 2013.