# Lower bounds on the chromatic number of random graphs

## TL;DR

This paper establishes a rigorous lower bound on the chromatic number of sparse random graphs using physics-inspired methods, improving understanding of graph coloring thresholds.

## Contribution

It provides the first rigorous proof of a physics-predicted lower bound on the chromatic number of sparse random graphs, using the interpolation method.

## Key findings

- Lower bounds match physics predictions for certain graph models
- Explicit bounds for small average degrees are derived
- Simplified derivation of asymptotic formulas for large degrees

## Abstract

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborova and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on the chromatic number of random graphs for small (average) degrees. Additionally, show how asymptotic formulas for large degrees that were previously obtained by lengthy and complicated combinatorial arguments can be re-derived easily from these new results.