Achromatic numbers for circulant graphs and digraphs

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that α (C16q2+20q+7(1, 2)) = 8q + 5, and for the diachromatic number we state that dac(C→\vec C32q2+24q+5(1, 2)) = 8q + 3. In general, we give the lower bounds α (C4q2+aq+1(1, a)) ≥ 4q + 1 and dac(C→\vec C8q2+2(a+4)q+a+3(1, a)) ≥ 4q + 3 when a is a non quadratic residue of ℤ4q+1 for graphs and ℤ4q+3 for digraphs, and the equality is attained, in both cases, for a = 3. Finally, we determine the achromatic index for circulant graphs of q2 +q + 1 vertices when the projective cyclic plane of odd order q exists.