When removing an independent set is optimal for reducing the chromatic number

TL;DR

This paper determines the precise threshold, in terms of maximum degree, at which removing an independent set is the most effective strategy to reduce a graph's chromatic number, using a reduction to Brooks' theorem.

Contribution

It provides a precise threshold for when removing an independent set optimally reduces the chromatic number based on maximum degree, refining previous understanding.

Findings

Threshold determined within two values for large maximum degree

Sometimes the threshold is uniquely identified

Results apply to graphs with sufficiently large maximum degree

Abstract

How large must the chromatic number of a graph be, in terms of the graph's maximum degree, to ensure that the most efficient way to reduce the chromatic number by removing vertices is to remove an independent set? By a reduction to a powerful, known stability form of Brooks' theorem, we answer this question precisely, determining the threshold to within two values (and indeed sometimes a unique value) for graphs of sufficiently large maximum degree.