Graph colourings may be viewed as special constraint satisfaction problems. The class of
-colouring problems enjoys a well known dichotomy of complexity — these problems are polynomial time solvable when
≤ 2, and NP-complete when
≥ 3. For general constraint satisfaction problems such dichotomy was conjectured by Feder and Vardi, but has still not been proved in full generality. We discuss some results and techniques related to this Dichotomy Conjecture. We focus on the effects of a new concept of ‘fullness’, and how it affects the complexity of constraint satisfaction problems and their dichotomy. Full constraint satisfaction problems may then be specialized back to graph colourings, yielding an interesting novel class of problems in graph theory, related to the study of graph perfection.