Independent sets in graphs with given minimum degree

TL;DR

This paper investigates the maximum number and size of independent sets in large graphs with a given minimum degree, proving a strengthened version of Galvin's conjecture about extremal graph structures.

Contribution

It establishes a strengthened form of Galvin's conjecture, identifying graphs that maximize independent sets in graphs with a specified minimum degree.

Findings

Identifies extremal graphs with maximum independent sets

Proves a strengthened version of Galvin's conjecture

Characterizes largest random independent sets in such graphs

Abstract

We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.