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

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