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arXiv · 2603.26214

Optimal b-Colourings and Fall Colourings in $H$-Free Graphs

Jungho Ahn·Tala Eagling-Vose·Felicia Lucke·David Manlove·Fabricio Mendoza·Daniël Paulusma
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Abstract

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in $H$-free graphs. For Tight b-Chromatic Number in $H$-free graphs, we develop a general technique to determine new graphs $H$, for which the problem is polynomial-time solvable, and we also determine new graphs $H$, for which the problem is still NP-complete. We show, for the first time, the existence of a graph $H$ such that in $H$-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

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Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, David Manlove, Fabricio Mendoza, Daniël Paulusma. 2026-03-27. Optimal b-Colourings and Fall Colourings in $H$-Free Graphs. https://arxiv.org/abs/2603.26214

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