Colouring graphs when the number of colours is almost the maximum degree

We consider the chromatic number of graphs with maximum degree Δ. For sufficiently large Δ, we determine the precise values of k   for which the barrier to (Δ+1−k)(Δ+1−k)-colourability must be a local condition, i.e. a small subgraph. We also show that for Δ constant and sufficiently large, (Δ+1−k)(Δ+1−k)-colourability is either NP-complete or can be solved in linear time, and we determine precisely which values of k correspond to each case.