Graphs with large girth are b-continuous

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 [χ(G),b(G)]. It is known that not all graphs are b-continuous. Here, we show that if G has girth at least 10, then G is b-continuous.