b-colouring outerplanar graphs with large girth

A b-colouring of a graph is a colouring of its vertices such that every colour class contains a vertex that has a neighbour in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b -colouring with k colours. We show here how to compute in polynomial time the b -chromatic number of an outerplanar graph of girth at least  8 . This generalizes the seminal result of Irving and Manlove on trees.