Graphs with Girth at Least 8 are b-continuous

Abstract

A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of G is the set S_b(G) of integers k such that G has a b-coloring with k colors and b(G) = maxS_b(G) is the b-chromatic number of G. A graph is b-continous if $S_b\left(G\right)=\left[\chi \right(G),b(G\left)\right]\cap Z$. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph G with girth at least 7 contains the integers between 2χ(G) and b(G). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.