We provide a polynomial-time algorithm for b -Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80(1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25(2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is \(\textsf{FPT}\) when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for \(b\)-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.

我们提供了一种多项式时间算法，用于在恒定团宽度的图上进行b 着色。这统一并扩展了几乎所有先前已知的图类多项式时间结果，并回答了 Campos 和 Silva (Algorithmica 80 (1), 104–115, 2018) 和 Bonomo 等人提出的开放性问题。（图与组合学25 (2), 153–167, 2009）。这是关于该问题的结构参数化的第一个结果。我们表明，当在一般图上通过顶点覆盖数参数化时，问题是\(\textsf{FPT}\) ，而在弦图上，当通过颜色数参数化时，问题是 \(\textsf{FPT}\) 。此外，我们观察到我们的有界团宽度图的算法可以适用于解决同一运行时范围内的秋季着色问题。在指数时间假设下，基于团宽度的\(b\) -着色和秋季着色算法的运行时间很紧。