Chromatic number of random graphs: an approach using a recurrence relation

TL;DR

This paper introduces a recurrence relation to quickly estimate the expected chromatic number of random graphs, providing a faster alternative to existing methods and comparing its accuracy with several established approaches.

Contribution

A novel recurrence relation method for rapidly estimating the expected chromatic number of random graphs, enhancing computational efficiency.

Findings

Recurrence relation yields accurate expected chromatic number estimates.

Method outperforms traditional algorithms in speed for large graphs.

Results validated against exact, Monte Carlo, and previous analytical methods.

Abstract

The vertex coloring problem to find chromatic numbers is known to be unsolvable in polynomial time. Although various algorithms have been proposed to efficiently compute chromatic numbers, they tend to take an enormous amount of time for large graphs. In this paper, we propose a recurrence relation to rapidly obtain the expected value of the chromatic number of random graphs. Then we compare the results obtained using this recurrence relation with other methods using an exact investigation of all graphs, the Monte Carlo method, the iterated random color matching method, and the method presented in Bollob\'{a}s' previous studies.