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.

翻译：在图染色中，若一个顶点与每种其他颜色的顶点均相邻，则称其为b-色顶点。我们考虑四个被广泛研究的染色问题：b-色数、严格b-色数、Fall色数和Fall全色数，它们可归入一个框架，该框架基于每个色类是否满足(i)至少有一个b-色顶点、(ii)恰有一个b-色顶点、或(iii)所有顶点均为b-色顶点。通过结合已知与新结果，我们完整分类了$H$-自由图中b-色数、Fall色数和Fall全色数的计算复杂性。针对$H$-自由图中的严格b-色数问题，我们发展了一种通用技术以确定新的图$H$，使该问题可在多项式时间内求解；同时我们也确定了新的图$H$，使该问题仍为NP-完全问题。我们首次证明了存在图$H$，使得在$H$-自由图中，b-色数为NP-困难问题，而严格b-色数可在多项式时间内求解。