The achromatic number of the union of cycles

The achromatic number of a graph G is the largest number of colors which can be assigned to the vertices of G so that adjacent vertices get different colors and each pair of distinct colors appears on the ends of some edge. We show that the achromatic number of the disjoint union of k cycles of length ℓ1,ℓ2,…,ℓk is equal to the achromatic number of the cycle of length p=∑i=1kℓi for any k⩽p/2, and that the achromatic number of the disjoint union of k triangles (resp. quadrangles) is equal to the achromatic number of the cycle of length 3k (resp. 4k) for any positive integer k.