Edge-b-coloring Trees

TL;DR

This paper presents a polynomial-time algorithm for b-coloring trees, specifically for claw-free block graphs, extending the class of graphs where the problem can be efficiently solved.

Contribution

The paper introduces a polynomial-time algorithm for b-coloring in claw-free block graphs, a subclass of chordal and line graphs, solving the problem for trees.

Findings

Polynomial-time algorithm for b-coloring claw-free block graphs

Extends efficient solutions to a broader class of graphs

Addresses the edge coloring version of the problem in trees

Abstract

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring with $b(G)$ colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing $b(G)$ is $\mathcal{NP}$-hard in general and polynomial-time solvable for trees. Since then, a number of complexity results were shown, including NP-hardness results for chordal graphs (Havet et. al., 2011) and line graphs (Campos et. al., 2015). In this article, we present a polynomial time algorithm that solves the problem restricted to claw-free block graphs, an important subclass of chordal graphs and line graphs. This is equivalent to solving the edge coloring version of the problem restricted to trees.