The achromatic number of a graph

The concept of coloring a graph has been shown to be subsumed by that of an homomorphism. This led in [3] to the definition of a complete n-coloring of a graph G and suggested therefore a new invariant, which we now call the “achromatic number” ψ(G). While the chromatic number χ(G) is the minimum number of colors required for (a complete coloring of) the points of G, the achromatic number is the maximum such number. We obtain several bounds for ψ(G) in terms of other invariants of a graph, and in particular we show that, for any graph G having p points, _x(G)+ͨ(G)¯⩽p+1, a result which generalizes a theorem of Nordhaus and Gaddum [4].