The b-chromatic number of a graph

The achromatic number psi(G) of a graph G=(V,E) is the maximum k such that V has a partition V-1, V-2,...,V-k into independent sets, the union of no pair of which is independent. Were we show that psi(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the b-chromatic number, phi(G), of G. We prove that determining phi(G) is NP-hard for general graphs, but polynomial-time solvable for trees. (C) 1999 Elsevier Science B.V. All rights reserved.