On the existence of graphs with prescribed coloring parameters

The pseudoachromatic number ψ(G) of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that between any two distinct parts, there is at least one edge of G. Here, we prove that if 2⩽a⩽b⩽c, then there exists a graph G with chromatic number a, achromatic number b and pseudoachromatic number c.