On the stability of graph independence number

TL;DR

This paper investigates the stability of the independence number in graphs, establishing bounds related to induced subgraphs and applying results to the Erdős–Rogers function.

Contribution

It provides a sharp upper bound on the independence number for graphs with stability conditions and derives new results for the Erdős–Rogers function.

Findings

Bound: lpha(G)  n/2 + C_{k, ll}

Sharpness for cases where k-ll  2

New values for the Erd
51s--Rogers function

Abstract

Let $G$ be a graph on $n$ vertices of independence number $\alpha(G)$ such that every induced subgraph of $G$ on $n-k$ vertices has an independent set of size at least $\alpha(G) - \ell$. What is the largest possible $\alpha(G)$ in terms of $n$ for fixed $k$ and $\ell$? We show that $\alpha(G) \le n/2 + C_{k, \ell}$, which is sharp for $k-\ell \le 2$. We also use this result to determine new values of the Erd\H{o}s--Rogers function.