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 alpha(C16q2+20q+7(1, 2)) = 8q+5, and for the diachromatic number we state that dac((C) over right arrow (32q2+24q+5)(1, 2)) = 8q + 3. In general, we give the lower bounds alpha(C4q2+aq+1(1, a)) >= 4q + 1 and dac((C) over right arrow (8q2+2(a+4)q+a+3)(1, a)) >= 4q+3 when a is a non quadratic residue of Z(4q+1) for graphs and Z(4q+3) for digraphs, and the equality is attained, in both cases, for a = 3. Finally, we determine the achromatic index for circulant graphs of q(2) + q + 1 vertices when the projective cyclic plane of odd order q exists.