Chromatic number of random graphs

Abstract

The vertex coloring problem is fundamental to various practical applications, including task scheduling and resource allocation. We investigate the expected value of the chromatic number for random graphs, which can be expressed as a polynomial in the number of vertices and the edge probability. We provide an exact polynomial expression for random graphs with 10 vertices. Furthermore, for random graphs with 11 vertices, we derive a polynomial using the Monte Carlo method. Our analysis indicates that this estimation maintains high numerical precision.