The b-continuity of graphs with large girth
  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. 
Ana Silva and Cl\'audia Linhares-Sales
math.CO cs.DM