A note on Two-Point Concentration of the Independence Number of $G_{n,m}$

TL;DR

This paper proves that the independence number of the random graph G_{n,m} is concentrated on two values in a specific edge regime, highlighting a difference from the G_{n,p} model where it varies with edge count.

Contribution

It establishes the two-point concentration of the independence number of G_{n,m} for certain m, revealing a fundamental difference from G_{n,p} in that regime.

Findings

Independence number of G_{n,m} is concentrated on two values for n^{5/4+ε} < m ≤ innom{n}{2}

Difference between G_{n,m} and G_{n,p} in independence number behavior in the specified regime

Variations in lpha(G_{n,p}) are driven by edge count fluctuations in that regime

Abstract

We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ \epsilon} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/4 + \epsilon} < m< n^{4/3}$. In this regime the independence number of $ G_{n,m}$ is concentrated on two values while the independence number of $ G_{n,p}$ is not; indeed, for $p$ in this regime variations in $ \alpha( G_{n,p})$ are determined by variations in the number of edges in $ G_{n,p}$.