b-coloring graphs with large girth

TL;DR

This paper presents a polynomial-time algorithm for computing the b-chromatic number of graphs with girth at least 9, extending previous results from trees to a broader class of graphs.

Contribution

It introduces a polynomial-time method for determining the b-chromatic number in graphs with large girth, improving upon prior work limited to trees.

Findings

Polynomial-time computation for girth ≥ 9

Extension of b-chromatic number results beyond trees

Improved understanding of graph coloring complexity

Abstract

A b-coloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. We show how to compute in polynomial time the b-chromatic number of a graph of girth at least 9. This improves the seminal result of Irving and Manlove on trees.