# Cliques and Chromatic Number in Inhomogenous Random Graphs

## TL;DR

This paper investigates the properties of cliques and chromatic numbers in inhomogeneous random graphs, providing bounds and estimates even when edge probabilities are very low, extending understanding of graph coloring in sparse and diverse networks.

## Contribution

It introduces a recursive method to estimate maximum clique size in inhomogeneous graphs and derives uniform bounds for both inhomogeneous and homogeneous cases across various edge probabilities.

## Key findings

- Maximum clique size estimates for inhomogeneous graphs.
- Uniform bounds on clique size and chromatic number for homogeneous graphs.
- Applicability across a wide range of edge probabilities.

## Abstract

In this paper, we study cliques and chromatic number of inhomogenous random graphs where the individual edge probabilities could be arbitrarily low. We use a recursive method to obtain estimates on the maximum clique size under a mild positive average edge density assumption. As a Corollary, we also obtain uniform bounds on the maximum clique size and chromatic number for homogenous random graphs for all ranges of the edge probability $p_n$ satisfying $\frac{1}{n^{\alpha_1}} \leq p_n \leq 1-\frac{1}{n^{\alpha_2}}$ for some positive constants $\alpha_1$ and $\alpha_2$.