The achromatic number of the union of paths Original Research Article

The achromatic number of a graph G is the largest number of colours which can be assigned to the vertices of G so that adjacent vertices get different colours and every pair of distinct colours appears on the ends of some edge. We consider the achromatic number of the disjoint union of paths of length a1,a2,…,ak, and show that it is the largest integer n such that a1+a2+⋯+ak⩾(n2)+f(k,n), where f(k,n) equals 0 if n is odd, or if n is even and k⩾n/2, and equals n/2−k if n is even and k