On the Achromatic Number of Hypercubes

The achromatic number of a finite graph G, ψ(G), is the maximum number of independent sets into which the vertex set may be partitioned, so that between any two parts there is at least one edge. For an m-dimensional hypercube Pm2 we prove that there exist constants 0<c1<c2, independent of m, such that c1(m2m−1)1/2⩽ψ(Pm2)⩽c2(m2m−1)1/2.