Proper colorings of a graph in linear time using a number of colors linear in the maximum degree of the graph

TL;DR

This paper introduces a novel linear-time algorithm for sampling proper graph colorings, efficient when the number of colors exceeds approximately 3.637 times the maximum degree, improving sampling speed for large graphs.

Contribution

The paper presents the first linear-time sampling algorithm for proper graph colorings under specific color-to-degree ratio conditions.

Findings

Expected running time is linear in graph size.

Algorithm works for color counts greater than 3.637 times the maximum degree.

First such efficient sampling algorithm for this problem.

Abstract

A new algorithm for exactly sampling from the set of proper colorings of a graph is presented. This is the first such algorithm that has an expected running time that is guaranteed to be linear in the size of a graph with maximum degree \( \Delta \) when the number of colors is greater than \( 3.637 \Delta + 1\).