Achromatic number of collections of paths and cycles

A complete colouring   of a simple graph GG is a proper vertex colouring such that each pair of colours appears together on at least one edge. The achromatic number  ψ(G)ψ(G) is the greatest number of colours in such a colouring.We give simple necessary and sufficient conditions for a graph of maximum degree 2 to have a complete colouring with kk colours, provided the graph is large enough, and use this to give the achromatic number for such a graph.