# Independent Sets in n-vertex k-chromatic, \ell-connected graphs

## TL;DR

This paper investigates the maximum number of independent sets in large, k-chromatic, -connected graphs, identifying extremal structures and extending results to graphs with minimum degree constraints and fixed independent set sizes.

## Contribution

It introduces new extremal results for the number of independent sets in k-chromatic, -connected graphs, including stability analysis and specific cases for fixed independent set sizes.

## Key findings

- Identified the unique extremal graph maximizing independent sets for large n.
- Extended results to graphs with minimum degree at least .
- Determined maximum independent sets of size 2 in k-chromatic, -connected graphs.

## Abstract

We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally address maximizing the number of independent sets of size 2 (equivalently, minimizing the number of edges) over all $n$-vertex $k$-chromatic $\ell$-connected graphs.