b-chromatic index of graphs

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to a vertex in each other color class. The b-chromatic number of G is the maximum integer χ b ( G ) for which G has a b-coloring with χ b ( G ) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χ b ( G ) is NP -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also NP -hard or not. Here, we prove that computing the b-chromatic index of a graph G is NP -hard, even if G is either a comparability graph or a C k -free graph, and give some partial results on the complexity of the problem restricted to trees.