  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.
