We show that the b-Coloring problem is complete for the class XNLP when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is W[t]-hard for all t, and resolves the parameterized complexity of b-Coloring parameterized by treewidth.
Cite
@article{arxiv.2209.07772,
title = {$b$-Coloring Parameterized by Pathwidth is XNLP-complete},
author = {Lars Jaffke and Paloma T. Lima and Roohani Sharma},
journal= {arXiv preprint arXiv:2209.07772},
year = {2022}
}