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

Cambie, Stijn and Haslegrave, John and Kang, Ross (2024) When removing an independent set is optimal for reducing the chromatic number. European Journal of Combinatorics, 115: 113781. ISSN 0195-6698

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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.