  In this work we show that, for any fixed d, random d-regular graphs
asymptotically almost surely can be coloured with k colours, where k is the
smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to
Molloy and Reed, this establishes the chromatic number to be asymptotically
almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is
discarded and thus the chromatic number is exactly determined. Hence we improve
a recently announced result by Achlioptas and Moore in which the chromatic
number was allowed to take the value k+1. Our proof applies the small subgraph
conditioning method to the number of balanced k-colourings, where a colouring
is balanced if the number of vertices of each colour is equal.
