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 χb(G) for which G has a b-coloring with χb(G) colors. This problem was introduced by Irving and Manlove in 1999, where they showed that computing χb(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 Ck-free graph, and give some partial results on the complexity of the problem restricted to trees.