The b-continuity of graphs with large girth

TL;DR

This paper proves that graphs with girth at least 10 are b-continuous, meaning they can be properly b-colored with any number of colors between their chromatic number and b-chromatic number.

Contribution

The paper establishes a new sufficient condition (girth at least 10) for graphs to be b-continuous, extending understanding of b-coloring properties.

Findings

Graphs with girth ≥ 10 are b-continuous.

Not all graphs are b-continuous, but high girth ensures this property.

The result broadens the class of graphs known to be b-continuous.

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 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. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[\chi(G),b(G)]$. It is known that not all graphs are b-continuous. In this article, we show that if $G$ has girth at least 10, then $G$ is b-continuous.