b-Coloring Parameterized by Clique-Width

TL;DR

This paper presents a polynomial-time algorithm for b-Coloring on graphs with bounded clique-width, unifying previous results and exploring structural parameterizations, including vertex cover and number of colors.

Contribution

It introduces the first structural parameterization results for b-Coloring and extends algorithms to Fall Coloring, with tight complexity bounds under ETH.

Findings

Polynomial-time algorithm for b-Coloring on graphs of constant clique-width

FPT algorithm for b-Coloring parameterized by vertex cover number

Algorithm for Fall Coloring with similar runtime bounds

Abstract

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, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is 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.