The b-chromatic Number of a Graph

The achromatic number (G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2, . . . , Vk into independent sets, the union of no pair of which is independent. Here we show that (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, (G), of G. We prove that determining (G) is NP-hard for general graphs, but polynomial-time solvable for trees.

• Type: Journal
• Year: 1999
• Where: DAM
• Authors: Robert W. Irving, David Manlove
• Keywords: Achromatic Number, DAM 1999, Independent Sets, Partial Order