Maximum independent sets on random regular graphs

TL;DR

This paper rigorously determines the asymptotic independence number of random d-regular graphs for all sufficiently large d, confirming heuristic predictions and providing explicit formulas for the main parameters.

Contribution

It provides a rigorous proof of the asymptotics of the independence number in random regular graphs, confirming the one-step replica symmetry breaking heuristics.

Findings

Asymptotic independence number characterized for all large d

Concentration results with explicit constants

Validation of replica symmetry breaking heuristics

Abstract

We determine the asymptotics of the independence number of the random $d$-regular graph for all $d \ge d_0$. It is highly concentrated, with constant-order fluctuations around $n\alpha_* - c_*\log n$ for explicit constants $\alpha_*(d)$ and $c_*(d)$. Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.