Achromatic number of collections of paths and cycles

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