$b$-Coloring Parameterized by Pathwidth is XNLP-complete

TL;DR

This paper establishes the exact complexity of the b-Coloring problem when parameterized by pathwidth, showing it is XNLP-complete and W[t]-hard for all t, thus resolving its complexity with respect to treewidth.

Contribution

It proves that b-Coloring parameterized by pathwidth is XNLP-complete and W[t]-hard for all t, clarifying its complexity landscape.

Findings

b-Coloring is XNLP-complete when parameterized by pathwidth.

The problem is W[t]-hard for all t with this parameterization.

It resolves the complexity status of b-Coloring parameterized by treewidth.

Abstract

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.