The achromatic number ψ( G) of a graph G = ( V, E) is the maximum k such that V has a partition V 1, V 2, h., V k into independent sets, the union of no pair of which is independent. Here we show that ψ( 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, ϑ( G), of G. We prove that determining ϑ( G) is NP-hard for general graphs, but polynomial-time solvable for trees.